Working with Vectors | Edexcel GCSE Maths: Foundation Revision Notes 2017

Finding Vector Paths

You need to be able to find the vector using a grid
- The grid will be made up of parallelograms or equilateral triangles
The following grid is made up entirely of parallelograms, with the vectors a and b defined as marked in the diagram:
- Any vector that goes horizontally to the right along a side of a parallelogram will be equal to a
- Any vector that goes up diagonally to the right along a side of a parallelogram will be equal to b

Vectors on a grid of parallelograms

To find the vector between two points
- Count how many times you need to go horizontally to the right
  - This will tell you how many a's are in your answer
- Count how many times you need to go up diagonally to the right
  - This will tell you how many b's are in your answer
- Add the a's and b's together
  - e.g. $\vec{A R} = 2 a + 3 b$
You will have to put a negative in front of the vector if it goes in the opposite direction
- -a is one length horizontally to the left
- -b is one length down diagonally to the left
  - e.g. $\vec{F B} = - b + a$

Vector paths on a grid

Adding and subtracting vectors follows all the same rules as adding and subtracting letters like a and b in algebra (this includes collecting like terms).
Always look for the easiest path between two points
- Go as far as you can in one direction
- And then use the other direction

The following diagram consists of a grid of identical parallelograms.

Vectors $a$ and $b$ are defined by $a = \vec{A B}$ and $b = \vec{A F}$ .

Vector parallelogram grid, IGCSE & GCSE Maths revision notes

Write the following vectors in terms of $a$ and $b$ .

\vec{A E}

To get from A to E we need to follow vector a four times to the right.

\begin{array}{rcl} \vec{A E} & = & \vec{A B} + \vec{B C} + \vec{C D} + \vec{D E} \\ = & a + a + a + a \end{array}

\vec{A E} = 4 a

\vec{G T}

There are many ways to get from G to T. One option is to go from G to Q (b twice), and then from Q to T (a three times).

\begin{array}{rcl} \vec{G T} & = & \vec{G L} + \vec{L Q} + \vec{Q R} + \vec{R S} + \vec{S T} \\ = & b + b + a + a + a \end{array}

\vec{G T} = 3 a + 2 b

\vec{E K}

There are many ways to get from E to K. One option is to go from E to O (b twice), and then from O to K ( -a four times).

\begin{array}{rcl} \vec{E K} & = & \vec{E J} + \vec{J O} + \vec{O N} + \vec{N M} + \vec{M L} + \vec{L K} \\ = & b + b - a - a - a - a \end{array}

\vec{E K} = 2 b - 4 a

$- 4 a + 2 b$ also acceptable