Vectors Notes

Q: How do I find the magnitude of a vector?

Use Pythagoras on the two components. The magnitude of a vector is its length. The horizontal and vertical components are the two shorter sides of a right-angled triangle and the vector is the hypotenuse, so |\mathbf{v}|=\sqrt{x^2+y^2}. Negative components still square to positive numbers.

Cambridge IGCSE Maths: Revision notes

Key facts

A vector has both magnitude and direction. Write it as a column $\binom{x}{y}$ , as $\overrightarrow{AB}$ or in bold as $\mathbf{a}$ .
Add, subtract and scale vectors component by component. A scalar multiple is parallel to the original.
Magnitude: $|\mathbf{v}|=\sqrt{x^2+y^2}$ .
$\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}$ (Extended).
Points are collinear if the vectors between them are parallel and share a common point.

What is a vector?

A vector is a quantity with both size and direction, drawn as an arrow.

A vector has magnitude (size) and direction. As a column vector $\binom{x}{y}$ it means $x$ units across and $y$ units up. $\overrightarrow{AB}$ is the vector from point $A$ to point $B$ . In print vectors are bold, like $\mathbf{a}$ , and by hand they are underlined. A vector is drawn as a directed line segment: an arrow showing length and direction.

Vector AB = (3, 2): 3 units right and 2 units up

Which of these is a vector quantity?

Adding, subtracting and scaling

Work with the top and bottom numbers separately; a scalar multiplies both.

To add or subtract vectors, add or subtract the matching components: $\binom23+\binom1{-4}=\binom3{-1}$ . To multiply by a scalar, multiply every component: $3\binom2{-1}=\binom6{-3}$ . The result is parallel to the original, longer or shorter, and reversed if the scalar is negative.

Adding vectors: go along a then b, or straight along a + b

Worked example

Given $\mathbf{a}=\binom21$ and $\mathbf{b}=\binom4{-3}$ , find $2\mathbf{a}+\mathbf{b}$ .

1
Scale first: $2\mathbf{a}=\binom42$ .
2
Add components: $\binom42+\binom4{-3}=\binom{4+4}{2-3}$ .

Work out $2\binom1{-3}+\binom41$ .

Magnitude

Use Pythagoras on the two components.

The magnitude of a vector is its length. The horizontal and vertical components are the two shorter sides of a right-angled triangle and the vector is the hypotenuse, so $|\mathbf{v}|=\sqrt{x^2+y^2}$ . Negative components still square to positive numbers.

The vector (3, 4) is the hypotenuse of a 3-4-5 right-angled triangle

Magnitude of $\binom xy$ $\sqrt{x^2+y^2}$

Worked example

Find the magnitude of $\binom34$ .

1
Square the components: $3^2+4^2=9+16=25$ .
2
Square root: $\sqrt{25}$ .

What is the magnitude of $\binom5{-12}$ ?

Position vectors

A position vector gives a point's place relative to the origin, and $\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}$ .

A position vector describes where a point is relative to the origin $O$ . The position vector of $A$ is $\overrightarrow{OA}$ . The vector between two points is the end minus the start. More generally, you can write any vector in a diagram in terms of two given vectors by tracing a route between the points.

Going from A to B by way of O: AB = −a + b = b − a

$\overrightarrow{AB}$ $\overrightarrow{OB}-\overrightarrow{OA}$

Worked example

$A$ has position vector $\binom21$ and $B$ has $\binom54$ . Find $\overrightarrow{AB}$ .

1
$\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}$ .
2
$\binom54-\binom21=\binom33$ .

$\overrightarrow{OA}=\binom12$ and $\overrightarrow{OB}=\binom46$ . Find $\overrightarrow{AB}$ .

Vectors in geometry

Parallel vectors are scalar multiples; collinear points need parallel vectors and a shared point.

Vectors prove facts without measuring. Lines are parallel if one vector is a scalar multiple of the other. Points $A$ , $B$ , $C$ are collinear if $\overrightarrow{AB}$ and $\overrightarrow{BC}$ are parallel and share the common point $B$ . If a point divides a line in a ratio, its position vector is a weighted combination of the end points.

Parallel lines

One vector is a scalar multiple of the other

Collinear points

Vectors are parallel
They share a common point

Worked example

Show that $A(1,2)$ , $B(3,5)$ and $C(7,11)$ are collinear.

1
$\overrightarrow{AB}=\binom23$ and $\overrightarrow{BC}=\binom46$ .
2
$\overrightarrow{BC}=2\overrightarrow{AB}$ , so they are parallel.
3
Both share point $B$ .

$\overrightarrow{PQ}=\binom21$ and $\overrightarrow{RS}=\binom63$ . Are PQ and RS parallel?

Try an exam question

$P$ is the point $(2,1)$ and $Q$ is the point $(6,4)$ . (a) Write $\overrightarrow{PQ}$ as a column vector. (b) Work out the magnitude of $\overrightarrow{PQ}$ . (c) Work out $2\overrightarrow{PQ}-\binom1{-2}$ .

[4 marks]

That's the notes covered.

Carry on to the next subtopic.