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Lesson overview

Geometry — vectors

A vector has both magnitude (length) and direction. Written as $\begin{pmatrix} a \\ b \end{pmatrix}$ or in bold: **u** = (a, b). Vectors can be added: $\begin{pmatrix} a \\ b \end{pmatrix} + \begin{pmatrix} c \\ d \end{pmatrix} = \begin{pmatrix} a + c \\ b + d \end{pmatrix}$ . Vectors can be multiplied by a scalar (number): $k\begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} ka \\ kb \end{pmatrix}$ . The magnitude (length) of a vector is $|\begin{pmatrix} a \\ b \end{pmatrix}| = \sqrt{a^2 + b^2}$ . Two vectors are parallel if one is a scalar multiple of the other. Vectors are used in geometry proofs and in modelling real-world situations like forces or displacements.

Adding vectors

Add the vectors $\vec{u} = \begin{pmatrix} 3 \\ 2 \end{pmatrix}$ and $\vec{v} = \begin{pmatrix} 1 \\ 4 \end{pmatrix}$ .

u then v, placed head-to-tail -- the resultant u+v goes straight from the start to the end.

Add the components separately. The triangle rule: place vectors head-to-tail to visualise the resultant.

Scalar multiplication

Find $2\vec{w}$ where $\vec{w} = \begin{pmatrix} -3 \\ 5 \end{pmatrix}$ .

2w points the same direction as w, but is twice as long.

Multiply each component by the scalar. The direction stays the same; the magnitude is scaled.

Finding the magnitude of a vector

Find the magnitude of $\vec{a} = \begin{pmatrix} 5 \\ 12 \end{pmatrix}$ .

Use Pythagoras: the magnitude is the hypotenuse of the right triangle with sides a and b.

Using vectors to find parallel lines

Are the vectors $\vec{p} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$ and $\vec{q} = \begin{pmatrix} 6 \\ 9 \end{pmatrix}$ parallel?

p and q point in exactly the same direction, just different lengths.

Vectors are parallel if one is a scalar multiple of the other. Check if each component has the same scale factor.

Worked example — vector geometry proof

ABCD is a parallelogram with $\vec{AB} = \begin{pmatrix} 4 \\ 2 \end{pmatrix}$ and $\vec{AD} = \begin{pmatrix} 1 \\ 3 \end{pmatrix}$ . Find $\vec{AC}$ and verify that diagonals bisect each other.

In a parallelogram ABCD, $\vec{AC}$ goes from A to C, which is the path from A to B plus B to C.
But BC is parallel and equal to AD, so $\vec{BC} = \vec{AD}$ .
Therefore, $\vec{AC} = \vec{AB} + \vec{AD}$ .
Calculate: $\vec{AC} = \begin{pmatrix} 4 \\ 2 \end{pmatrix} + \begin{pmatrix} 1 \\ 3 \end{pmatrix} = \begin{pmatrix} 5 \\ 5 \end{pmatrix}$ .
Similarly, $\vec{BD} = \vec{AD} - \vec{AB}$ (from B to A, then A to D).
The diagonals intersect at their midpoints, confirming the parallelogram property.

Try it

Work with the components carefully. Remember: add components separately for addition, multiply each component for scalar multiplication.

Question 1

Add $\begin{pmatrix} 2 \\ 5 \end{pmatrix} + \begin{pmatrix} 3 \\ -1 \end{pmatrix}$ .

💡 Add the first components: 2 + 3 = 5. Add the second: 5 + (−1) = 4.

Question 2

Find $3\begin{pmatrix} 2 \\ -4 \end{pmatrix}$ .

💡 Multiply each component by 3: $\begin{pmatrix} 3 × 2 \\ 3 × (-4) \end{pmatrix}$ .

Question 3

Find the magnitude of $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$ .

💡 $|\vec{v}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$ .

Question 4

Are $\begin{pmatrix} 2 \\ 5 \end{pmatrix}$ and $\begin{pmatrix} 4 \\ 10 \end{pmatrix}$ parallel?

💡 $\begin{pmatrix} 4 \\ 10 \end{pmatrix} = 2\begin{pmatrix} 2 \\ 5 \end{pmatrix}$ , so they are parallel.

Question 5

Find $\begin{pmatrix} 6 \\ 2 \end{pmatrix} - \begin{pmatrix} 1 \\ 5 \end{pmatrix}$ .

💡 Subtract components: 6 − 1 = 5, and 2 − 5 = −3.

Common mistakes

Watch for these when working through the lesson.

Forgetting to add both components when adding vectors. Add x-components and y-components separately.
Not multiplying both components when doing scalar multiplication.
Confusing vector subtraction with addition. Remember: $\vec{a} - \vec{b} = \vec{a} + (-\vec{b})$ .
Using the wrong formula for magnitude. Always use $|\vec{v}| = \sqrt{a^2 + b^2}$ (Pythagoras), not just $a + b$ .

Vectors: addition, scalar multiplication and magnitude

Lesson overview

Worked example — vector geometry proof

Try it

Common mistakes

Related topics

Independent practice will plug in here