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Vectors I

Representing vectors

Representing vectors

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Further kinematics

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Variable acceleration in one dimension

Differentiating vectors

Integrating vectors

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Maxima and minima problems

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Constant acceleration

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Constant acceleration: Deriving formulae

Constant acceleration formulae: SUVAT

Vertical motion under gravity

Modelling in mechanics

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Quantities and units in mechanics

Working with vectors

Normal distribution

The normal distribution

The normal distribution: Finding probabilities

The inverse normal distribution function

The standard normal distribution

Finding the mean and standard deviation

Normal approximation to the binomial distribution

Hypothesis testing with the normal distribution

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Hypothesis testing I

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Vectors II

3D coordinates

Vectors in 3D

Solving geometric problems

Applications to mechanics: 3D vectors

Numerical methods

Locating roots

Iteration

Staircase and cobweb diagrams

Newton-Raphson method

Numerical methods: Applications to modelling

Integration II

Integrating standard functions

Integrating f(ax + b)

Integration with trigonometric identities

Reverse chain rule

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Finding areas using integration

The trapezium rule

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Differentiation II

Differentiating sin x and cos x

Differentiating exponentials and logarithmic functions

The chain rule

The product rule

The quotient rule

Differentiating inverse functions

Differentiating trigonometric functions

Parametric differentiation

Implicit differentiation

Second derivatives: Concave and convex functions

Connected rates of change

Trigonometry and modelling

Addition formulae

Using the angle addition formulae

Double angle formulae

Solving trigonometric equations: Double angle formula

Simplifying a cos x + b sin x

Proving trigonometric identities

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Further trigonometric identities

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Radians

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Solving trigonometric equations

Small angle approximations

Sequences and series

Sequences revision

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Arithmetic series

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Geometric series

Sum to infinity

Sigma notation

Modelling with sequences and series

Binomial expansion II

Binomial expansion: (1+x)^n

Binomial expansion: (a+bx)^n

Binomial expansion with partial fractions

Binomial expansion of compound expressions

Parametric equations

Parametric equations

Using trigonometric identities

Sketching parametric curves

Points of intersection of parametric curves

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Functions

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The modulus function

y = |f(x)| and y = f(|x|)

Combining transformations

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Modulus inequalities

Algebraic fractions

Operations with algebraic fractions

Partial fractions

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Algebraic division

Proof II

Proof by contradiction

Criticising proofs

Vectors I

Vectors

Representing vectors

Magnitude and direction of a vector

Position vectors and displacement vectors

Solving geometric problems

Modelling with vectors

Integration I

Integrating x^n

Indefinite integrals

Finding functions by integrating

Definite integrals

Area under a curve

Area under the x-axis

Area between a curve and a line

Differentiation I

Finding the gradient of a curve

Differentiation from first principles

Differentiating x^n

Differentiating functions with multiple terms

Gradients, tangents and normals

Increasing and decreasing functions

Second order derivatives

Stationary points

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Exponentials and logarithms

Exponential functions

y = e^x

Exponential modelling

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Natural logarithms

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Angles in all four quadrants

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Equations and identities

Trigonometry

The cosine rule

The sine rule

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Trig graphs: sin x, cos x, tan x

Transforming trigonometric graphs

Binomial expansion I

Pascal's triangle

Factorial notation

Binomial expansion

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Binomial estimation

Circles

Midpoints and perpendicular bisectors

Equation of a circle

Intersection of straight lines and circles

Using tangent and chord properties

Circles and triangles

Intersection of two circles

Straight line graphs

y = mx + c

Equations of straight lines

Parallel and perpendicular lines

Length and area

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Transformations

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Points of intersection

Using the discriminant

Direct and inverse proportion

Polynomials

Functions

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The factor theorem

Equations and inequalities

Linear simultaneous equations

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Set notation and interval notation

Linear inequalities

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Inequality regions

Quadratics

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Identifying the discriminant

Solving disguised quadratics

Indices and surds

Laws of indices

Negative and fractional indices

Surds

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Proof I

Mathematical structure and arguments

Disproof by counterexample

Proof by deduction

Proof by exhaustion

Explainer Video

Tutor: Alice

Summary

​​Representing vectors

In a nutshell

You can represent vectors in column vector form (ab)\begin{pmatrix}a\\b\end{pmatrix}(ab​). This shows its displacement relative to the xxx- and yyy- axes. You can add or subtract column vectors and multiply vectors by a scalar. A unit vector is a vector with a magnitude of 1. Unit vectors in the positive xxx​ and yyy​ directions are called 'i\textbf{i}i' and 'j\textbf{j}j', repesectively.


Definitions

Column vector

A vector in column forming, showing displacement relative to the xxx- and yyy-axes. (ab)\begin{pmatrix}a\\b\end{pmatrix}(ab​) means move aaa steps right and bbb steps up.​

Unit vector

A vector with a magnitude of 111. Unit vectors in the positive xxx​ and yyy​ directions are called 'i\textbf{i}i​' and 'j\textbf{j}j​', repesectively.

Component form

Writing a vector using i\textbf{i}i and j\textbf{j}j notation.



Column vectors

A vector describes a displacement between points. In a two-coordinate system, you can write this displacement relative to the xxx- and yyy-axes in column vector form. v=(ab)\textbf{v}=\begin{pmatrix}a\\b\end{pmatrix}v=(ab​) means move aaa units along the positive xxx​-axis, then bbb​ units along the positive yyy​-axis.​


Example 1

Write a\textbf{a}a​ in column vector form.

Maths; Vectors I; KS5 Year 12; Representing vectors

The vectors takes 222 steps right then 333 steps up, so:


a=(23)‾\textbf{a}=\underline{\begin{pmatrix}2\\3\end{pmatrix}}a=(23​)​​​

​

​

Adding and subtracting vectors

To add and subtract column vectors, work out the operations row by row:


​(ab)+(cd)=(a+cb+d)\boxed{\begin{pmatrix}a\\b\end{pmatrix}+\begin{pmatrix}c\\d\end{pmatrix}=\begin{pmatrix}a+c\\b+d\end{pmatrix}}(ab​)+(cd​)=(a+cb+d​)​​​



Multiplying a vector by a scalar

To multiply a vector by a scalar kkk, multiply each term by the scalar:


​k(ab)=(kakb)\boxed{k\begin{pmatrix}a\\b\end{pmatrix}=\begin{pmatrix}ka\\kb\end{pmatrix}}k(ab​)=(kakb​)​​​


Example 2

If a=(21)\textbf{a}=\begin{pmatrix}2\\1\end{pmatrix}a=(21​) and b=(31)\textbf{b}=\begin{pmatrix}3\\1\end{pmatrix}b=(31​), find a+b\textbf{a}+\textbf{b}a+b.


​a+b=(21)+(31)=(2+31+1)=(52)‾\begin{aligned}\textbf{a}+\textbf{b}=\begin{pmatrix}2\\1\end{pmatrix}+\begin{pmatrix}3\\1\end{pmatrix}=\begin{pmatrix}2+3\\1+1\end{pmatrix}=\underline{\begin{pmatrix}5\\2\end{pmatrix}}\end{aligned}a+b=(21​)+(31​)=(2+31+1​)=(52​)​​​​​


Example 3

If a=(21)\textbf{a}=\begin{pmatrix}2\\1\end{pmatrix}a=(21​), find −3a-3\textbf{a}−3a.

−3a=−3(21)=(−3×2−3×1)=(−6−3)‾-3\textbf{a}=-3\begin{pmatrix}2\\1\end{pmatrix}=\begin{pmatrix}-3\times2\\-3\times1\end{pmatrix}=\underline{\begin{pmatrix}-6\\-3\end{pmatrix}}−3a=−3(21​)=(−3×2−3×1​)=(−6−3​)​​​



Unit vectors

A unit vector is a vector with a magnitude of 111​. For the positive xxx and yyy directions, the unit vectors are commonly referred to as i\textbf{i}i and j\textbf{j}j, respectively.​


​​i=(10)j=(01)\boxed{\begin{aligned}\textbf{i}&=\begin{pmatrix}1\\0\end{pmatrix}\\\\\textbf{j}&=\begin{pmatrix}0\\1\end{pmatrix}\end{aligned}}ij​=(10​)=(01​)​​​​
Maths; Vectors I; KS5 Year 12; Representing vectors

​

Any two-dimensional vector v\textbf{v}v can be written in terms of i\textbf{i}i​ and j\textbf{j}j, or component form:

v=(ab)=ai+bj\boxed{\textbf{v}=\begin{pmatrix}a\\b\end{pmatrix}=a\textbf{i}+b\textbf{j}}v=(ab​)=ai+bj​​​

​

Example 4

Draw a diagram to represent the vector 4i−3j4\textbf{i}-3\textbf{j}4i−3j and rewrite it in column vector form.

The diagram would be:

Maths; Vectors I; KS5 Year 12; Representing vectors

In column vector form:

4i−3j=(4−3)‾4\textbf{i}-3\textbf{j}=\underline{\begin{pmatrix}4\\-3\end{pmatrix}}4i−3j=(4−3​)​​​


Example 5

If a=2i+5j\textbf{a}=2\textbf{i}+5\textbf{j}a=2i+5j and b=i−3j\textbf{b}=\textbf{i}-3\textbf{j}b=i−3j, find a+2b\textbf{a}+2\textbf{b}a+2b, giving your answer in terms of i\textbf{i}i and j\textbf{j}j.​


a+2b=(2i+5j)+2(i−3j)=(2i+5j)+(2i−6j)=4i−j‾\begin{aligned}\textbf{a}+2\textbf{b}&=(2\textbf{i}+5\textbf{j})+2(\textbf{i}-3\textbf{j})\\&=(2\textbf{i}+5\textbf{j})+(2\textbf{i}-6\textbf{j})\\&=\underline{4\textbf{i}-\textbf{j}}\end{aligned}a+2b​=(2i+5j)+2(i−3j)=(2i+5j)+(2i−6j)=4i−j​​​​


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FAQs - Frequently Asked Questions

What is a unit vector?

A vector with a magnitude of 1.

How do you add or subtract column vectors?

To add and subtract column vectors, add and subtract each row individually.

What is a column vector?

A vector in column form, showing displacement relative to the x- and y- axes.

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