Representing Vectors as Diagrams (Cambridge (CIE) IGCSE International Maths)

Revision Note

Written by: Jamie Wood

Reviewed by: Dan Finlay

Vector Diagrams

How can I represent a vector visually?

A vector has both a size (magnitude) and a direction
- You need to draw a line to show the size of the vector
- You also need to draw an arrow to show the direction of the vector

Vectors are written in bold when typed to show that they are a vector and not a scalar
- When writing a vector in an exam you should underline the letter to show it is a vector
- $a$ when typed and $a$ when handwritten
  - You will not lose marks if you forget to underline vectors

If a vector starts at A and ends at B we can write it as $\vec{A B}$
- Here the arrow will point toward B
- Vector $\vec{B A}$ will have the same length but point toward A

How do I draw a vector on a grid?

You can draw a vector anywhere on a grid
- Just make sure it has the correct length and the correct direction
To draw the vector $a = (\begin{matrix} 3 \\ 4 \end{matrix})$
- Pick a point on the grid and draw a dot there
- Count 3 units to the right and 4 units up and draw another dot
- Draw a line between the two dots
- Put an arrow on the line pointing toward the second dot
Look out for negatives and zeroes
- $b = (\begin{matrix} 2 \\ - 4 \end{matrix})$ goes 2 to the right and 4 down
- $c = (\begin{matrix} 2 \\ 0 \end{matrix})$ goes 2 to the right but does not go up or down

What happens when I multiply a vector by a scalar?

When you multiply a vector by a positive scalar:
- The direction stays the same
- The length of the vector is multiplied by the scalar
For example, $a = (\begin{matrix} 4 \\ - 2 \end{matrix})$
- $2 a = (\begin{matrix} 8 \\ - 4 \end{matrix})$ will have the same direction but double the length
- $\frac{1}{2} a = (\begin{matrix} 2 \\ - 1 \end{matrix})$ will have the same direction but half the length

When you multiply a vector by a negative scalar:
- The direction is reversed
- The length of the vector is multiplied by the number after the negative sign
For example, $a = (\begin{matrix} 4 \\ - 2 \end{matrix})$
- $- a = (\begin{matrix} - 4 \\ 2 \end{matrix})$ will be in the opposite direction and its length will be the same
- $- 2 a = (\begin{matrix} - 8 \\ 4 \end{matrix})$ will be in the opposite direction and its length will be doubled

Multiplying a vector by a negative scalar

What happens when I add or subtract vectors?

To draw the vector $a + b$
- Draw the vector $a$
- Draw the vector $b$ starting at the endpoint of $a$
- Draw a line that starts at the start of $a$ and ends at the end of $b$
To draw the vector $a - b$
- Draw the vector $a$
- Draw the vector $- b$ starting at the endpoint of $a$
- Draw a line that starts at the start of $a$ and ends at the end of $- b$

Worked Example

The points A, B and C are shown on the following coordinate grid.

(a)

Write the vectors $\vec{A B}, \vec{A C}$ and $\vec{C B}$ as column vectors.

Start by drawing the three vectors onto the grid

Question points with vectors, IGCSE & GCSE Maths revision notes

From A to B, it is 6 to the right and 2 up

$\vec{A B} = (\begin{matrix} 6 \\ 2 \end{matrix})$

From A to C, it is 7 to the right and 6 down

$\vec{A C} = (\begin{matrix} 7 \\ - 6 \end{matrix})$

From C to B, it is 1 to the left and 8 up

$\vec{C B} = (\begin{matrix} - 1 \\ 8 \end{matrix})$

(b)

Without using any calculations, explain why $\vec{A B} + \vec{B C} + \vec{C A} = (\begin{matrix} 0 \\ 0 \end{matrix})$ .

The vector goes from A to B, then from B to C, then from C back to A

The vector returns to its starting point

Last updated: 7 October 2024

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