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- Maths and statistics
- Functions and graphs
Absolute values turn negative values into positive ones, and leave positive ones unchanged. Absolute value functions are useful for expressing situations where only magnitude matters, without regard to direction, like measuring distance, temperature differences, profit or loss diffeerences, and margin of error. Absolute values The absolute value of any number...
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- Maths and statistics
- Vectors and matrices
- Matrices
The addition and subtraction of matrices is important across many applications, from computer graphics and image processing to statistics. Use this resource to learn the rules for adding and subtracting matrices. Addition of matrices There is one key rule for adding matrices. Matrices of the same shape (same number of...
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- Maths and statistics
- Integration
Antidifferentiation is the reverse operation of differentiation. You need it to solve problems involving area under a curve, total accumulated change, and reversing rates of change. By mastering antidifferentiation, you'll be able to reconstruct original functions and solve real-world problems in physics, engineering, and beyond. If you have the derivative...
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- Maths and statistics
- Integration
Integrals can be used to find the area between a curve and the \(x\)-axis. We can use this to calculate the total accumulation of a quantity in physics, economics and engineering. Use this resource to learn how to calculate area under the curve. Finding the area under a curve Consider...
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- Maths and statistics
- Numbers and measurement
Arithmetic forms the foundation of mathematics, focusing on the basic operations and properties of numbers. You need arithmetic to perform calculations and solve everyday problems. Use this resource to enhance your numerical skills. Numbers Fractions Decimals Percentages...
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- Maths and statistics
- Vectors and matrices
- Matrices
Matrices can be used to solve systems of equations using elementary row operations and the augmented matrix. This method of solving systems of equations is handy when you have many variables and equations. You may see it being used in fields like economics, statistics and machine learning. Systems of equations...
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- Maths and statistics
- Statistics and probability
Binomial probability is used in various scenarios, like quality control and genetics. By understanding binomial probability, we can make informed predictions and decisions based on repeated experiments. Use this resource to learn how to apply the concepts of binomial probability. Binomial probability ("bi" meaning "two" and "nomos" meaning "parts") helps...
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- Maths and statistics
- Functions and graphs
Circular functions produce periodic wave-like graphs and are foundational in trigonometry. They have broad applications across physics, engineering, and signal processing, where modelling repetitive phenomena and wave patterns is essential. Use this resource to explore circular functions. Circular functions include sine, cosine, and tangent. They describe relationships involving angles and...
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- Maths and statistics
- Trigonometry
Trigonometric functions such as sin, cos and tan are usually defined as the ratios of sides in a right-angled triangle. These ratios can be extended to angles greater than \(90^{\circ}\), using angles in a unit circle. Circular functions have applications in many STEM disciplines, such modelling oscillating motion in engineering,...
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- Maths and statistics
- Algebra
Complex numbers are a group of numbers that help us to get mathematical solutions where real numbers (which include positive and negative, counting numbers, fractions and decimals) just can’t work. Use this resource to get started with complex numbers. Introduction to complex numbers...
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- Maths and statistics
- Statistics and probability
If two events are not independent, then the outcome of one event can change the probability of the second event occurring. Understanding the connection between these helps us make better predictions and decisions based on specific situations. Use this resource to learn about the concept of conditional probability. Conditional probability...
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- Maths and statistics
- Statistics and probability
When we use the mean of a sample to estimate the mean of an entire population, we use confidence intervals to help us understand the precision of our estimates and how confident we can be about them. This tool is useful for making informed decisions based on sample data. Use...
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- Maths and statistics
- Differentiation
You can sketch an accurate graph of a function if you know some of its key characteristics. Use this resource to learn how to bring together everything to sketch a curve. To sketch a curve, it is helpful to know the key points of the function, like: maxima, minima, or...
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- Maths and statistics
- Statistics and probability
Data is everywhere. It drives many aspects of our day-to-day lives. Use this resource to learn about the different types of data and some ways of displaying it. Collecting and processing data To study the relationships between variables (a characteristic, number, or quantity that can be measured or counted), you...
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- Maths and statistics
- Numbers and measurement
- Arithmetic
Decimals are used to express fractions in a more precise and easily understandable form. They are essential for handling money, measuring quantities, and applying mathematical concepts in everyday life and various professional fields. Use this resource to learn how to add, subtract, multiply, divide and convert decimals. A decimal is...
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- Maths and statistics
- Hyperbolic functions
The differentiation and integration of hyperbolic functions allow us to understand how these functions change and accumulate values which teaches us the essential tools for analysing complex mathematical and real-world systems. Keep reading to learn about these functions and put your new knowledge to the test with some exercises. See...
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- Maths and statistics
- Vectors and matrices
- Matrices
A determinant is a number that can be calculated for any square matrix. The determinant is used in calculating vector cross products, eigenvalues, eigenvectors and solving simultaneous equations. Use this resource to learn how to find the determinant of \(2\times2\) and \(3\times3\) matrices. Determinant of a \(2\times2\) matrix The determinant...
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Differential equations help us capture the relationships between changing variables. They provide a framework for analysing how different factors interact over time and let us to predict future behavior. Use this resource to learn how to solve differential equations. Introduction to differential equations First order separable First order linear Second...
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Differentiation is a fundamental concept in calculus that deals with finding the rate at which things change. It is used to calculate slopes of curves, optimise functions in economics, and model motion in physics. Differentiation helps us understand how quantities vary, from tracking speed to finding maximum profit. Use this...
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- Maths and statistics
- Differentiation
Learn how to take a derivative of a function using first principles. Using this method is the best way to understand the concepts around differentiation. Derivative of a function The derivative of a function \(f(x)\) is denoted by \(f'(x)\). It is defined as: \[f'(x)=\lim_{h\rightarrow0}\left(\frac{f(x+h)-f(x)}{h}\right)\quad h\neq0\] Using this definition is called...
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- Maths and statistics
- Numbers and measurement
- Measurement
Our everyday world is filled with things we can measure; each is defined by dimensions in maths and physics. By understanding dimensions, we can accurately describe and compare different physical quantities. Use this resource to learn about dimensions. Dimensions describe the fundamental units that make up physical properties. When we...
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- Maths and statistics
- Vectors and matrices
- Vectors
Imagine navigating a hilly landscape, searching for the steepest ascent or smoothest descent—this is the essence of directional derivatives. These mathematical tools are vital in meteorology for predicting temperature changes along wind paths, in finance for analysing portfolio shifts, and in machine learning for optimising algorithms. Use this resource to...
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- Maths and statistics
- Vectors and matrices
- Vectors
The distance between a point in 3D space and a plane can be determined using vectors and trigonometry. This is handy in many situations, like making sure components are specific distances from certain planes in construction, determining distances between a airplane and a mountain or building in safe flight planning,...
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- Maths and statistics
- Integration
What if you have an area that is bounded by limits on both the \(x\) and the \(y\)-axes? This is where double integrals come into play. We use double integrals in physics for finding mass in a region given a density function, engineering for determining properties of materials with varying...
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- Maths and statistics
- Vectors and matrices
- Matrices
Eigenvalues and eigenvectors are important parts of an engineer’s mathematical toolbox. They give us an understanding of how buildings, structures, automobiles and materials behave in real life. But that's not all they are used for; you can see their use in many other areas of STEM, including colour theory, electric...
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- Maths and statistics
- Vectors and matrices
- Vectors
Uniquely defining a line for a vector in three-dimensional space is useful in a range of scenarios. In mechanical and civil engineering, they are needed to create models and analyse alignment. In computer graphics, they can be used to define paths for animations. In physics, they represent trajectories of particles...
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- Maths and statistics
- Vectors and matrices
- Vectors
Just as we can define an equation for a line in three dimensions, we can do the same for an entire plane–that is, we can define an equation that represents a two-dimensional space within a 3D space. A plane is a subset of three dimensional space. You can think of...
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- Maths and statistics
- Algebra
Solving algebraic equations involves finding the values of variables that satisfy mathematical statements. Build on your foundational understanding of algebra by tackling how to solve basic equations. Solving linear equations Solving simultaneous equations...
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- Maths and statistics
- Differential equations
First order linear differential equations are are widely used in various fields, such as physics, engineering, and biology, to model processes like cooling, population growth, and electrical circuits. Understanding how to solve first order linear differential equations helps you analyse and predict the behavior of systems in the real world....
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- Maths and statistics
- Differential equations
The \(x\) and \(y\) terms in first order separable differential equations can be separated to make them easier to solve. These equations are seen when modelling population growth or chemical reactions. Use this resource to learn how to solve them. First order separable differential equations A first order separable differential...
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- Maths and statistics
- Numbers and measurement
- Indices, surds and logarithms
An index can be an integer – a counting number - either positive or negative. It can also be a fraction such as \(\dfrac{1}{2}\), \(\dfrac{3}{4}\), or \(2.5\). Use this resource to learn how the laws of indices apply to fractional indices. Index laws Remember the basic index laws: \(a^{m}\times a^{n}=a^{m+n}\)...
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- Maths and statistics
- Numbers and measurement
- Arithmetic
Mastering fractions is crucial for tackling a wide range of mathematical problems. The skill is applicable to a huge range of real-world contexts, from cooking and construction to science and finance. Use this resource to learn how to simplify, add, subtract, multiply and divide fractions. A fraction is the easiest...
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Maths uses many symbols, or notations, which represent different functions and operations. You will come across lots of these as you explore the Learning Lab. Use this glossary to learn what any unfamiliar symbols mean. Foundational notations Foundational notations appear across different areas of maths and science. Symbol Explanation \(+\)...
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- Maths and statistics
- Differentiation
Gradients and tangents help us understand how functions change. We use derivatives to find these. Learn how these concepts work together to analyse and interpret the behaviour of functions. Tangent at a point along a curve A tangent is a line that touches a curve at only one point. Where...
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- Maths and statistics
- Functions and graphs
Simple functions and relations can be transformed into more complicated functions. Seeing how changes in equations affect their visual representation will give you a deeper understanding of functions. This is useful in modelling real-world situations and solving complex maths problems. Use this resource to learn about transformations of graphs. Summary...
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- Maths and statistics
- Differentiation
Higher order derivatives reveal even deeper insights into the behaviour of functions, such as jerk in motion or the curvature of graphs. These concepts are important in fields like physics, engineering and economics, where understanding complex dynamics is crucial. Higher order derivatives are derivatives of derivatives, providing insights into how...
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- Maths and statistics
- Functions and graphs
Hybrid functions combine different types of functions. They are useful for modelling real-world scenarios that have different behaviours under different conditions, like tax brackets or shipping rates. Other examples include analysing frictional forces in physics, designing profiles and shapes for vehicles, and modelling free-fall acceleration. Use this resource to learn...
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- Maths and statistics
- Hyperbolic functions
The hyperbolic functions are analogous to the circular (trigonometric) functions and are widely used in engineering, science and mathematics. This module introduces hyperbolic functions, their graphs and similarities to the circular functions. Whereas circular functions are defined on a unit circle, the hyperbolic functions are defined on a hyperbola. Hyperbolic...
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Information about functions and graphs to improve your maths skills in these areas. HF1 Hyperbolic functions This module introduces hyperbolic functions, their graphs and similarities to the circular functions. HF2 Derivatives and integrals of hyperbolic functions The hyperbolic functions are widely used in engineering, science and mathematics. HF3 Inverse hyperbolic...
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- Maths and statistics
- Statistics and probability
Hypothesis testing is used anywhere we conduct research to determine whether there is a relationship between variables. These days, hypothesis testing forms the basis of machine learning and data science, where it is used to validate models and make sure they reliably detect patterns and make predictions. Use this resource...
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- Maths and statistics
- Differentiation
Implicit differentiation enables you to find the derivative of y with respect to x without having to solve the original equation for y. Implicit functions Sometimes, equations involve variables that are intertwined and not easily separated. For example, \(y^{5}+3xy+x^{2}-5=0\) or \(y=\sin(xy))\). In such expressions, \(y\) is said to be an...
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- Maths and statistics
- Numbers and measurement
- Indices, surds and logarithms
Index notation is a powerful mathematical tool for expressing repeated multiplication concisely. Learn how to work with indices, understand the laws governing them, and convert expressions involving powers and roots. This will help you simplify complex expressions and solve problems across a wide range of mathematical and scientific disciplines. Video...
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- Maths and statistics
- Numbers and measurement
Indices, surds and logarithms extend our understanding of numbers and their relationships. Use these resources to enhance your comprehension of these powerful mathematical tools. Indices Surds Fractional indices Logarithms...
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Integration is a fundamental concept in calculus that involves finding the total accumulated change or area under a curve. It is used to calculate distances from velocity, determine areas and volumes, and solve differential equations. Integration helps us understand the cumulative effects of changing quantities, from determining total growth to...
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- Maths and statistics
- Integration
Integration by parts is used to solve integrals by expressing them in terms of simpler integrals. This is applied in areas like mechanical engineering for calculating work done by variable forces, in physics for solving complex integrals, and in economics for evaluating growth models. Use this resource to learn how...
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- Maths and statistics
- Integration
Integration by substitution simplifies the process of finding integrals by making a change of variables. This is used for calculating areas and volumes in engineering, solving differential equations in physics, and optimising functions in economics. Use this resource to learn how to integrate by substitution. An expression composed of two...
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- Maths and statistics
- Integration
How do you integrate an exponential function? This skill is important for calculating compound interest over time in finance, determining population growth models in biology, and analysing radioactive decay in physics. Use this resource to learn how. Exponential functions have the form: \[y=ae^{kx}\] where \(a\) and \(k\) are constants. Examples...
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- Maths and statistics
- Integration
How do you integrate polynomial functions, where different terms have different powers and coefficients? This skill is important for calculating projectile trajectories in physics, beam deflections in engineering and determining cost functions and profit maximisation in economics. Use this resource to learn how. In Antidifferentiation, you learned how to find...
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- Maths and statistics
- Integration
How do you integrate a reciprocal function? This skill is important for calculating electrical currents in circuits and modelling harmonic motion in mechanical systems. Use this resource to learn how. Reciprocal functions have the form: \[y=\frac{k}{x}\] where \(k\) is a constant. We often deal with reciprocal functions when looking at...
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- Maths and statistics
- Integration
How do you integrate a trigonometic function? This skill is important for solving problems involving wave patterns in physics, analysing alternating current circuits in engineering and determining structural loads with curved geomoetries in architecture. Use this resource to learn how. You already learnt about the general forms of Circular functions....
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- Maths and statistics
- Integration
How do you integrate an expression when there is an algebraic expression in the numerator and denominator of a fraction? Integrating using partial fractions helps you to solve this problem. Use this resource to learn how to integrate using partial fractions. Sometimes a complex function may be integrated by breaking...
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- Maths and statistics
- Vectors and matrices
- Vectors
Vectors can be used to determine whether two lines in 3D cross each other (or intersect), and identify the point at which they intersect. This is used in a variety of STEM disciplines, including detection of collisions between objects in robotics and modelling complex structures in computer graphics and game...
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- Maths and statistics
- Vectors and matrices
- Vectors
Two planes in 3D can intersect. Finding this intersection has many real-life applications, including the design of buildings in architecture, 3D rendering and modelling in computer graphics, and determining paths of movement in robotics. Use this resource to learn how to determine the angle between two intersecting planes and the...
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- Maths and statistics
- Functions and graphs
Interval notation helps to clearly express a continuous range of numbers in a simple, concise way. It is handy in many ways, such as to define acceptable weight limits for a machine, indicating a valid range of measurements in scientific experiments, and specifying temperature ranges in which a machine is...
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- Maths and statistics
- Differential equations
Differential equations help us understand and predict behaviours in STEM, where they are used to model phenomena like motion, heat transfer, population dynamics, and financial markets. Use this resource to learn about first order separable differential equations. Differential equations A differential equation is an equation that includes a function and...
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- Maths and statistics
- Vectors and matrices
- Matrices
Use this resource to learn about matrices, their order, and the different types of matrices. A sound understanding of matrices is essential for a wide range of real-world applications, like machine learning, natural language processing, robotics, image processing, urban planning and infrastructure development. Matrices Matrices are arrays of numbers or...
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- Maths and statistics
- Vectors and matrices
- Vectors
Vectors are used to represent forces in physics, handle 2D and 3D manipulation with computer graphics, and to calculate the forces acting on materials in textiles. You also deal with vector quantities in your everyday life, from driving your car down the road to planning the shortest route to get...
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- Maths and statistics
- Functions and graphs
One-to-one functions can be inverted so that the inputs and outputs are reversed. They are crucial where we need effective reversal of processes and calculations, like undoing equations in physics, computer graphics transformations, or decrypting data. Use this resource to learn about inverse functions and how to graph them. If...
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- Maths and statistics
- Hyperbolic functions
Providing a function is one to one, it is possible to find an inverse function. This module discusses inverse hyperbolic functions, which are used in advanced calculus for integration and in the solution of differential equations. See Hyperbolic functions for a list of definitions. Inverse hyperbolic functions may also be...
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- Maths and statistics
- Vectors and matrices
- Matrices
In matrix algebra, we can add, subtract and multiply matrices as long as the matrix order is correct. Unlike traditional arithmetic, we cannot divide matrices. Instead, we multiply by the inverse matrix. Inverse matrices have many applications, including computer animation, encryption and digital image transformations. Inverse matrices An inverse matrix...
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- Maths and statistics
- Vectors and matrices
- Matrices
Being able to find the inverse of a \(3\times3\) matrix will help to simplify complex problems and enhances your ability to perform matrix operations efficiently. This is crucial in fields like engineering, physics and computer science. Before you read further, make sure that you are familiar with augmented matrices and...
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Transforms are another means of solving some differential equations that may prove too difficult to solve using other methods. Download the laplace transforms worksheets to improve your skills in these areas. LT1 Basic definition of laplace transforms (PDF) LT2 Table of transforms (PDF) LT3 Solving differential equations (PDF) LT4 Convolutional...
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- Maths and statistics
- Differentiation
Limits help us explore how functions behave as they approach specific points or infinity, providing the foundation for concepts like continuity and derivatives. You need this skill in many real-world applications, from engineering to physics, where predicting behaviour is crucial. Use this resource to learn how to calculate and interpret...
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- Maths and statistics
- Differentiation
We can use derivatives to estimate how a function behaves around a certain point. With linear approximation, we can analyse these small variations and predict values. These methods are useful in fields like physics and engineering, where precise measurements and predictions are crucial, but often challenging to obtain directly. Sometimes...
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- Maths and statistics
- Functions and graphs
Linear functions are the most basic type of function. They are used extensively across STEM and other disciplines, especially to model the rate of physical, biological or economic change. Use this resource to learn about linear functions and graphs. Linear functions create a straight line when graphed. They have the...
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- Maths and statistics
- Differentiation
Logarithmic differentiation makes it easier to differentiate complex functions. This method is particularly useful for functions with variable exponents or intricate products. You might encounter logarithmic differentiation when calculating compound interest, analysing population growth models or optimising processes in engineering and physics. Sometimes it is easier to differentiate the logarithm...
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- Maths and statistics
- Numbers and measurement
- Indices, surds and logarithms
The modeling of growth and decay in areas such as finance, epidemiology and science makes use of equations with logarithms and exponentials. The laws for working with logarithms enable us to solve equations that cannot be solved with other algebraic techniques. Use this resource to learn about them. Video tutorial...
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Further resources...
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- Maths and statistics
- Vectors and matrices
Matrices are used in a wide range of fields and applications to represent and manipulate data in a structured way. Begin your journey with these resources. Introduction to matrices Addition and subtraction of matrices Matrix multiplication Determinant of a matrix...
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- Maths and statistics
- Vectors and matrices
- Matrices
A strong understanding of how to multiply matrices is critical in many fields, like computer science, robotics, statistics and economics. Just as there are rules for adding and subtracting matrices, we can multiply matrices with each other when certain requirements are met. Use this resource to learn how to multiply...
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- Maths and statistics
- Differentiation
Taking the derivative of the function lets us find critical points where the slope is \(0\) or undefined. These points tell us where the function might have a maximum or a minimum value. This approach is useful in many real-world situations, like optimising resources or finding the best design for...
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- Maths and statistics
- Statistics and probability
The mean, mode, and median provide a single value that is typical of the data. These measures of central tendency help summarise large datasets, making it easier to understand the overall pattern. Understanding these concepts is essential for analysing and interpreting data effectively. Measures of central tendency The mean, median,...
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- Maths and statistics
- Numbers and measurement
Measurement forms the basis of many practical applications in STEM fields. Use this resource to develop a solid understanding of fundamental measurement concepts. Working through these pages will equip you with the skills you need for further exploration in science, technology, engineering, and mathematics. Units and conversions Dimensions Notation...
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- Maths and statistics
- Statistics and probability
By looking at how data in a set is distributed, we can see how much the data varies. This helps us understand the data better and make good decisions based on it. Use this resource to learn about the measures of spread. The measure of spread shows how spread out...
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- Maths and statistics
- Numerical methods
You are used to solving equations using basic algebraic operations and perhaps the quadratic formula. However, some equations cannot be solved using these methods. For example the equations: \[\begin{align*} e^{-x} & =x & \left(1\right) \end{align*}\] or \[\begin{align*} -\ln\left(x\right) & =x^{3} & \left(2\right) \end{align*}\] cannot be solved using conventional methods. However...
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- Maths and statistics
- Numerical methods
Some integrals cannot be evaluated in terms of the rules of integration or elementary functions. Simpson’s rule is a numerical method that calculates a numerical value for a definite integral. You will have evaluated definite integrals such as \[\begin{align*} \int_{1}^{3} & \left(x^{2}\right)dx \end{align*}\] before. In doing this, you are evaluating...
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- Maths and statistics
- Numerical methods
The trapezoidal rule is a numerical method that calculates a numerical value for a definite integral. Some integrals cannot be evaluated in terms of the rules of integration or elementary functions. You will have evaluated definite integrals such as \[\begin{align*} \int_{1}^{3} & \left(x^{2}\right)dx \end{align*}\] before. In doing this, you are...
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- Maths and statistics
- Statistics and probability
If data is normally distributed, it is represented by a bell-shaped curve. Normal distributions are common in nature and statistics, like heights or test scores. Understanding this distribution helps us make predictions and compare data from different sources easily. Graphical representations of data may look quite different. Many things that...
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- Maths and statistics
- Numbers and measurement
- Measurement
Imagine trying to write out the mass of the Earth in kilograms; that would be \(5\,972\,000\,000\,000\,000\,000\,000\,000\textrm{ kg}\). Expressing such big numbers this way is hard and can lead to mistakes. In science and engineering, notation plays a key role in simplifying calculations and conveying information efficiently. Use this resource to...
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- Maths and statistics
- Numbers and measurement
- Arithmetic
Learn the basics for working with integers, including negative numbers. Numerical reasoning skills you will learn here are very fundamental to maths; you will apply them to everything from everyday situations to more complex mathematical challenges. Integers Counting or whole numbers, also known as integers, can be used in many...
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Numerical methods are a collection of techniques used to solve mathematical problems that are difficult or impossible to solve with exact solutions. These worksheets are filled with information that will improve your skills in numerical methods. NM1 Newton’s method Some equations cannot be solved using algebra or other mathematical techniques....
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Further resources Also, the Learning Lab has sections that will help nursing students with important academic concepts:...
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- Maths and statistics
- Statistics and probability
One-tailed hypothesis tests are useful when you have a clear idea about the direction of an effect, such as testing if a new drug performs better than the standard treatment, whether a new fabric treatment increases the durability of a textile, or if a new alloy used for aircrafts is...
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- Maths and statistics
- Statistics and probability
As well as comparing the test statistic to a critical value, we can test hypotheses by comparing a p-value to the significance level, \(\alpha\). Understanding p-values helps us decide if our results are meaningful or just due to random chance. Use this resource to learn how to use p-values. In...
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- Maths and statistics
- Differentiation
Partial derivatives help us analyse the influence of each individual variable on a system. This is crucial for understanding surfaces and optimisation problems in fields like engineering, physics and economics. Use this resource to learn how to find partial derivatives. Partial derivatives reveal how a function with many variables changes...
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- Maths and statistics
- Numbers and measurement
- Arithmetic
Percentages are a powerful tool for expressing proportions, comparisons and changes. Learn how to convert fractions and decimals into percentages using this resource. This will enhance your ability to interpret data and make informed decisions when faced with quantities. A percentage is another way to express a part of a...
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An object in motion tends to stay in motion and an object at rest tends to stay at rest unless acted upon by an unbalanced force. Force equals mass times acceleration. For every action there is an equal and opposite reaction. Do you need to learn about Newton's first three...
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We will look at the gravitational force acting on an object on a slope. These can be divided into two components, the normal (resisting) force pushing into the slope which produces friction and the shear or driving force pushing the block down the slope. So we must consider forces parallel...
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Momentum is a vector quantity, so it is important to remember that direction must be taken into account when doing problems on momentum. Let's explore how we can do this. The moment p of an object is the product of its mass \(m\) and velocity \(v\), or: \[\begin{align*} p &...
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- Maths and statistics
- Functions and graphs
Plotting points on a graph is a foundational skill in maths. Mastering this technique enables you to visualise mathematical relationships and solve equations effectively. Use this resource to get started. The Cartesian plane The Cartesian plane has two perpendicular axes which intersect (cross over) at the origin \(O\). The horizontal...
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Linear motion refers to the motion of an object in a straight line. Describing these motions require some technical terms such as displacement, distance, velocity, speed and acceleration. The terms and their relationships to one another are described in this module. Scalar and vector quantities Quantities that have only a...
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- Maths and statistics
- Statistics and probability
The Poisson distribution deals with the number of random occurrences over a period of time or a given space, such as the number of people who enter a shop every hour or the number of flaws in a sheet of glass. It is very handy, well beyond applications in STEM....
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- Maths and statistics
- Statistics and probability
To find probabilities in normal distributions that aren't standard, we need to adjust the data to a common scale using \(z\)-scores. This allows us to find the likelihood of certain outcomes in a range of contexts, like identifying manufacturing tolerances, testing the tensile strength of new materials or calculating the...
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- Maths and statistics
- Statistics and probability
Probability rules help us understand how likely different outcomes are. They are applied in a range of ways, like risk assessment and modelling. Having a good understanding of these basics will help you to predict and make sense of different possibilities. When considering a set of all possible outcomes, an...
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- Maths and statistics
- Vectors and matrices
- Vectors
You will have learned that vectors can be resolved in two dimensions along the horizontal and vertical axes. It is also possible to resolve one vector along the line of another vector, instead of along the \(x\)- and \(y\)-axes. Often, in physics, engineering and mathematics courses, you are asked to...
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- Maths and statistics
- Numbers and measurement
- Reporting measurements
When multiple measurements are used to calculate a quantity, each of these measurements may carry its own uncertainty. By being careful and taking into account these uncertainties, scientists and engineers can better assess the reliability and precision of their conclusions. Use this resource to learn how to consider, or propagate,...
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- Maths and statistics
- Trigonometry
Pythagoras’ theorem shows the relationship between the sides of a right-angled triangle. Knowing the length of two sides of a right-angled triangle, the length of the third side can be calculated. Understanding Pythagoras' theorem helps you solve problems in fields such as architecture, engineering, and computer graphs, where precise calculations...
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- Maths and statistics
- Algebra
- Equation solving
The quadratic formula provides a straightforward way to find the solutions of any quadratic equation, especially when factoring is difficult or impossible. By becoming familiar with this formula, you'll gain a reliable tool for solving quadratic equations and deepen your understanding of algebraic relationships. Solving quadratic equations The solutions to...
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- Maths and statistics
- Functions and graphs
Quadratic functions have a distinct parabolic shape. Graphs of quadratic functions have applications in a wide range of fields, like modelling projectile motion, optimising business revenue, analysing population growth and designing bridges and arches. Use this resource to learn about parabolas and how to sketch them. A quadratic function has...
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- Maths and statistics
- Numbers and measurement
- Reporting measurements
The quality of measurements is very important in science and engineering because it affects how reliable the results are. There are a number of factors that affect the quality of measurements, like accuracy, precision, uncertainty and error. Use this resource to learn about these concepts. Accuracy and precision Accuracy shows...
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- Maths and statistics
- Differentiation
If there is a relationship between two or more variables, then there will also be a relationship between how these variables change. You may need to find how fast one variable changes in relation to another variable that is changing. This is called the rate of change. We often study...
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- Maths and statistics
- Functions and graphs
Understanding relations and functions is crucial for exploring more advanced topics in math and their applications in various scientific and engineering areas. Use this resouce to learn about what functions and relations are. Relations A relation is a set of ordered pairs. For example, \((1,2),(2,6),(3,4),(x,y)\) are ordered pairs. If the...
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- Maths and statistics
- Vectors and matrices
- Vectors
Breaking vectors down into their components—or resolving them—makes it easier to add or subtract them, especially when dealing with vectors that don't act along the same line. You will encounter this in many areas of STEM, like when analysing forces involved in robotics, studying the projectile motion of objects launched...
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- Maths and statistics
- Trigonometry
Explore the relationships between the angles and side lengths of a right-angled triangle. You will learn how to use trigonometric ratios to find unknown angles and lengths, enhancing your ability to solve geometric problems. Mastering this concept is vital for fields such as engineering, physics, and architecture, where precise calculations...
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- Maths and statistics
- Differentiation
Operational rules for differentiation allow us to handle complex expressions by breaking them down into simpler parts. They are used in real-world applications, such as calculating the speed of a moving car or determining the growth rate of a population. Understanding these rules makes it easier to calculate derivatives in...
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- Maths and statistics
- Statistics and probability
A sample space lists of all the possible outcomes, giving a complete picture of the situation. Sample spaces form the foundation for calculating probabilities and making predictions about future events. Use this resource to learn some ways to represent a sample space. A sample space is a list or diagram...
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- Maths and statistics
- Statistics and probability
Sampling distributions help us understand the behaviour of sample statistics, like means or proportions, from different samples of the same population. By examining these distributions, we can see how sample results might vary and how close they are likely to be to the actual population value. This concept is important...
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- Maths and statistics
- Vectors and matrices
- Vectors
There are two ways to multiply two vectors. Here, we will learn about the scalar product. It has many applications in STEM. For example, scalar products are used to calculate the work done by a system, in computer graphics to calculate the amount of light hitting surfaces, and in engineering...
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- Maths and statistics
- Differential equations
Second order homogeneous differential equations are crucial for modelling natural phenomena, like mechanical vibrations, electrical circuits and wave propagation. By learning how to solve these equations, you can understand and predict how systems behave under various conditions. Second order homogeneous differential equations A second order homogeneous differential equation is an...
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- Maths and statistics
- Differential equations
Second order non-homogeneous differential equations are used to model systems like climate dynamics with variable weather patterns, economic models with fluctuating market forces, and chemical reactions with external catalysts. Understanding how to solve them helps predict system responses to external stimuli. Second order non-homogeneous differential equations A second order non-homogeneous...
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- Maths and statistics
- Numbers and measurement
- Reporting measurements
In our everyday world, we measure things with precision. This is where significant figures come in. They help us communicate how precise our measurements are. Use this resource to learn how to identify and apply significant figures. A significant figure (or "sig fig") is a digit in a measurement that...
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- Maths and statistics
- Numbers and measurement
- Reporting measurements
Significant figures are important in calculations because they help keep the precision of measurements accurate in the results. When you do math operations, the number of significant figures in your values decides how precise the final answer should be. Use this resource to learn the rules for using significant figures...
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- Maths and statistics
- Algebra
- Equation solving
Learn how to solve cubic equations, where the highest power of the variable is three. These types of equations can model various real-world scenarios, from physics to engineering. The factor theorem The long division of polynomials is an important tool for solving cubic equations. However, the factor theorem is equally...
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- Maths and statistics
- Algebra
- Equation solving
Exponential equations are common in fields such as science, finance, and population dynamics, where exponential growth or decay is observed. By learning how to isolate variables and find solutions to exponential equations, you will be able to handle a variety of real-world problems involving exponential relationships. Exponential equations Indicial (or...
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- Maths and statistics
- Algebra
- Equation solving
Earlier, we looked at how to transpose formulas. Now, let's combine those skills with arithmetic to solve linear equations. By mastering the techniques for solving these equations, you'll be equipped to tackle a wide range of problems and build a strong basis for more advanced mathematical concepts. Video tutorial –...
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- Maths and statistics
- Algebra
- Equation solving
Just as linear equations can be solved to find the value of an unknown variable, we can do the same with quadratic equations. Here, we will learn how to solve quadratic equations using the null factor law. By applying this principle, you can factorise quadratic equations and determine their solutions...
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- Maths and statistics
- Algebra
- Equation solving
Delve into solving simultaneous equations, where a solution works for multiple equations. This skill is essential for analysing systems with multiple variables and is widely used in fields such as engineering, economics, and science. Simultaneous equations Simultaneous equations are equations that share variables and must be solved at the same...
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- Maths and statistics
- Trigonometry
A calculator may only give you one answer to an inverse trigonometry question between \(0\) and \(90^{\circ}\). When you use the unit circle, you can visualise the many different solutions for the angle. Understanding this concept will help you in many STEM fields, such as satellite navigation in physics, and...
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- Maths and statistics
- Vectors and matrices
- Matrices
There are a number of important special matrices to become familiar with if you are studying a STEM discipline. The transpose of a matrix is important for solving problems in structural mechanics, symmetric matrices can be applied to motion, dynamics and force calculations in physics, and orthogonal matrices have many...
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- Maths and statistics
- Statistics and probability
A standard normal distribution helps us compare different datasets by converting them to a common scale using values called \(z\)-scores. Understanding standard normal distributions makes it easier to calculate probabilities and assess how individual data points relate to the average. Use this resource to learn how to analyse standard normal...
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- Maths and statistics
- Statistics and probability
You may have seen the Greek symbol sigma (\(\Sigma\)) used in maths, but what does it mean? It is a way to simply express long numbers, using something called summation notation. A good understanding of summation notation is important in statistics for calculating averages and sums. Use this resource to...
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- Maths and statistics
- Numbers and measurement
- Indices, surds and logarithms
How do you find a square root of any number that isn’t already a square? The square root of \(16\) is \(4\) and the square root of \(25\) is \(5\), but can you find the square root of a number between these, for example \(\sqrt{20}\)? Learning how to identify and...
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- Maths and statistics
- Statistics and probability
T-tests help us understand if the differences we observe in data are due to random chance or if they're meaningful. They are commonly used in research to compare groups, like testing if a new teaching method improves scores more than the old one. Use this resource to learn how to...
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- Maths and statistics
- Statistics and probability
Rather than comparing a sample mean to a population mean, we can compare a sample proportion to a population proportion. These tests of proportion are used broadly, like for checking if a new fabric meets quality standards by having fewer defects or evaluating customer satisfaction levels for financial services. Use...
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- Maths and statistics
- Differentiation
The chain rule helps us differentiate larger functions composed of smaller functions. Use this resource to learn how to apply the chain rule. Video tutorial – using the chain rule Watch this video to learn how to use the chain rule to find the derivative of a function. The chain...
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- Maths and statistics
- Trigonometry
Like the sine rule, the cosine rule comes in handy for triangles that are not right-angled. Use this resource to learn about the cosine rule, how it is applied to find unknown sides and angles, and its importance in solving complex geometric problems involving any type of triangle. The cosine...
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- Maths and statistics
- Differentiation
The product rule helps us differentiate functions that are one function multiplied by another. Use this resource to learn how to apply the product rule. Video tutorial – using the product rule Watch this video to learn how to use the product rule to find the derivative of a function....
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- Maths and statistics
- Differentiation
The quotient rule helps us differentiate functions that are one function divided by another. Use this resource to learn how to apply the quotient rule. Video tutorial – using the quotient rule Watch this video to learn how to use the quotient rule to find the derivative of a function....
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- Maths and statistics
- Trigonometry
The trigonometric ratios are handy for real-life situations where right-angled triangles are involved, but what about triangles that are not right-angled? We use the sine rule. Use this resource to learn what the sine rule is and how it is applied. The sine rule shows that the ratio of the...
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- Maths and statistics
- Trigonometry
Angles are commonly measured in degrees but sometimes, it is useful to define them in terms of the length around the unit circle. This resource introduces radians as a measure of angle. Being able to use radians is crucial for solving problems in engineering, such as analysing the motion of...
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- Maths and statistics
- Algebra
- Algebra
Expand your transposition skills by exploring how to rearrange formulas involving brackets and fractions. By doing this, you will develop the skills needed to isolate variables in more complex expressions and prepare yourself well for advanced problem-solving in STEM. You will have learned the necessary theory when you were first...
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- Maths and statistics
- Trigonometry
Trigonometric identities are essential tools for simplifying expressions and solving equations in mathematics. They are used in a wide range of disciplines, including architecture and engineering, physics, computer graphics, astronomy and medical imaging. Use this resource to learn about some of the fundamental identities, double angle formulas, sums and difference,...
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- Maths and statistics
- Vectors and matrices
- Vectors
The vector product is another way to multiply two vectors. Just like scalar products, vector products have many broad applications, such as in electrical engineering, quantum physics, software development, game programming and statistics. Use this resource to learn more. The vector product is a vector resulting from multiplying the magnitudes...
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- Maths and statistics
- Vectors and matrices
Vectors are quantities that have both magnitude (size) and direction. This branch of maths is fundamental to physics and engineering to represent physical quantities that have a direction. Use this resource to get the basics down. Introduction to vectors Resolution of vectors Scalar or dot product Projection of vectors...
