Modulus of a Vector (Edexcel IGCSE Maths B): Revision Note

If there is no diagram, sketch one!

You can sketch a vector and use it to form a right-angled triangle.

Consider two points $A\left(-3,5\right)$ and $B\left(7,1\right)$.

(a) Write down the column vector $\overrightarrow{AB}$.

Answer:

Find the horizontal and vertical distances between the two points
Subtract the x and y components of A from B

$\overrightarrow{AB}=\left(\begin{array}{c}7--3\\ 1-5\end{array}\right)$

$\overrightarrow{AB}=\left(\begin{array}{c}10\\ -4\end{array}\right)$

(b) Find the modulus of vector $\overrightarrow{AB}$.

Answer:

Sketching a diagram of the vector $\overrightarrow{AB}$ can help

Apply Pythagoras' theorem to the x and y components of $\overrightarrow{AB}$

$\begin{array}{rcl}\left|\overrightarrow{AB}\right|& =& \sqrt{{10}^2+{\left(-4\right)}^2}\\ & =& \sqrt{100+16}\\ & =& \sqrt{116}\end{array}$

$\begin{array}{rcl}\mathit{\left|}\overrightarrow{AB}\mathit{\right|}& =& 2\sqrt{29}\end{array}$

(c) Briefly explain why $\left|\overrightarrow{BA}\right|=\left|\overrightarrow{AB}\right|$.

Answer:

The modulus of a vector is it's 'size'

Direction of the vector is ignored

$\left|\overrightarrow{BA}\right|=\left|\overrightarrow{AB}\right|$ since both vectors have the same length

(d) Another vector, $\overrightarrow{CD}$, has three times the modulus of vector $\overrightarrow{AB}$.

Write down a possible column vector for $\overrightarrow{CD}$.

Answer:

Being three times $\left|\overrightarrow{AB}\right|$ means the vector $\overrightarrow{AB}$ is three times longer

One way to find a vector is to multiply each component of the vector $\overrightarrow{AB}$ by 3 or -3

$\begin{array}{rcl}\overrightarrow{CD}& =& 3\overrightarrow{AB}=3\left(\begin{array}{c}10\\ -4\end{array}\right)\end{array}$

$\overrightarrow{CD}=\left(\begin{array}{c}30\\ -12\end{array}\right)$

Another possible answer is $\overrightarrow{CD}=\left(\begin{array}{c}-30\\ 12\end{array}\right)$