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Vectors Revision


Vectors are quantities that are defined by both magnitude and direction. They are often used to describe where points are, or how to get from one corner of a shape to another.

You will need to use and understand vectors, as well as be able to add, subtract, and multiply them.

Vector Notation

There are several ways a vector can be represented, so it’s important you are familiar with them all to prepare for your exams.

Bold / Underlined

The most common way you’ll see is a letter written in bold, for example, these two lines are vectors:

For example, the vector \boldsymbol{b} is in the direction of the pink arrow.

Note: when you are writing in your exam, as you can’t write in bold, you will be expected to underline vectors, like: \underline{a}

Column Notation

Vectors may also be displayed in columns:

Example: The column vector \begin{pmatrix}2 \\ 1\end{pmatrix} means 2 spaces to the right and 1 space up.

If you had \begin{pmatrix}-2 \\ -1\end{pmatrix}, this would mean 2 spaces to the left and 1 space down.


Finally, you may see vectors written like:

\overrightarrow{AB} which is the vector between point A and point B

\overrightarrow{OC} which is the vector between the origin and point C

Note: The point O almost always represents the origin in vector questions.

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Adding and Subtracting Vectors

When adding and subtracting using diagrams, it is important to pay attention to the direction the vector arrow is going.

A vector \boldsymbol{a} is positive in one direction, and if you flip the direction (point the arrow towards the opposite direction), the vector is now \boldsymbol{-a}:


If we have two vectors, \boldsymbol{a} and \boldsymbol{b}, these can be added:


\boldsymbol{a}+\boldsymbol{b} will follow a route from the start of the \boldsymbol{a} arrow, to the end of the \boldsymbol{b} arrow, both in the positive direction:


For subtraction, for example,


The \boldsymbol{b} is negative, so either of these diagrams correctly portray this subtraction:


Adding and Subtracting Column Vectors

Column vectors are easy to add and subtract, because you just treat the top row and the bottom row as separate sums, for example:

\begin{pmatrix}5 \\ 2\end{pmatrix}+\begin{pmatrix}3 \\ 1\end{pmatrix}=\begin{pmatrix}5+3 \\ 2+1\end{pmatrix}=\begin{pmatrix}8 \\ 3\end{pmatrix}

\begin{pmatrix}6 \\ 4\end{pmatrix}-\begin{pmatrix}2 \\ 3\end{pmatrix}=\begin{pmatrix}6-2 \\ 4-3\end{pmatrix}=\begin{pmatrix}4 \\ 1\end{pmatrix}

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Multiplying Vectors by a Scalar

Multiplying vectors by a scalar simply means multiplying vectors by a number (not a vector).


For example,


This is represented in the diagram:


If we have a vector containing more than one letter, we just multiply each letter by the scalar:


Similarly, in column vector form:

4\times\begin{pmatrix}3 \\ 2\end{pmatrix}=\begin{pmatrix}4\times3 \\ 4\times2\end{pmatrix}=\begin{pmatrix}12 \\ 8\end{pmatrix}

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Vector Magnitudes

The magnitude of a vector is the length of a vector.

Magnitude has no direction, so is always positive.



Find the magnitude of the vector

\overrightarrow{XY}=\begin{pmatrix}3 \\ 4\end{pmatrix}

Looking at the diagram, we can work out the length of the vector \overrightarrow{XY} by treating it as a right-angled triangle and using Pythagoras’ theorem:


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Example 1: Finding Vectors

Express the vector \overrightarrow{XZ} in terms of \boldsymbol{a} and \boldsymbol{b}

[2 marks]

So, let’s start at X. We will have to go through Y to get to Z.

To get from X to Y, we have to go in the negative direction of \boldsymbol{b}, so


Then, we will have to go from Y to Z, which is \boldsymbol{a} in the positive direction,


So, in total:




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Example 2: Vectors on Parallel Lines

Express the vectors \overrightarrow{AB} and \overrightarrow{AD} in terms of \boldsymbol{a} and \boldsymbol{b} in this parallelogram.

[2 marks]

With vectors, parallel lines of the same length are equal, as they have the same direction and magnitude.

So, in this parallelogram \overrightarrow{AB}  is equal to \overrightarrow{DC}, and \overrightarrow{AD} is equal to \overrightarrow{BC}.

So, to find \overrightarrow{AB}

We know this is the same as \overrightarrow{DC}, which, in this direction, equals \boldsymbol{-b}. So,


And as \overrightarrow{AD} is equal to \overrightarrow{BC},



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Example 3: Using Ratio

ABC is a triangle.

M lies on the line AB such that AM:MB is 1:1

Express the vector AM in terms of \boldsymbol{a} and \boldsymbol{b} 

[3 marks]

The ratio 1:1 means M is the midpoint of \overrightarrow{AB}.

Firstly, finding \overrightarrow{AB},


\overrightarrow{AM} is half of \overrightarrow{AB}, so,



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Vectors Example Questions








=2\times (-2\boldsymbol{a-b})





=-2\boldsymbol{a-b} - (3\boldsymbol{a-b})



Using Pythagoras,


As this is a regular hexagon, \overrightarrow{FA} and \overrightarrow{DC} are parallel, \overrightarrow{AB} and \overrightarrow{ED} are parallel, and \overrightarrow{EF}  and \overrightarrow{CB} are parallel.

To find \overrightarrow{FB}, we need \overrightarrow{FA} plus \overrightarrow{AB}.

\overrightarrow{FA} is the parallel to \overrightarrow{DC}, but in the opposite direction, so,



To find \overrightarrow{AB} , we can start by working out \overrightarrow{OC} which is parallel.

\overrightarrow{OC}=-3\boldsymbol{a}-\boldsymbol{b} -(5\boldsymbol{b}+\boldsymbol{b})=-8\boldsymbol{a}-2\boldsymbol{b}






Let’s firstly find \overrightarrow{AC} 




As \overrightarrow{AM}:\overrightarrow{MC} is 2:1, \overrightarrow{MC} is \dfrac{1}{3}\overrightarrow{AC}


\overrightarrow{AC}=\dfrac{1}{3}\times (9\boldsymbol{a}-3\boldsymbol{b})


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