Working with vectors in two and three dimensions: vector components, adding and subtracting vectors, multiplying by a scalar, and calculating the magnitude of a vector.

What this dot point is asking

The SQA wants you to work with vectors written in component form, add and subtract them, multiply a vector by a scalar (a number), and calculate the magnitude (length) of a vector in both two and three dimensions.

Vectors in component form

A vector describes a movement: so many across, so many up (and so many in or out, in three dimensions). It is written as a column of components. The vector $\begin{pmatrix} 3 \\ 2 \end{pmatrix}$ means "$3$ in the $x$ direction and $2$ in the $y$ direction". A vector has a size and a direction but no fixed position, so two arrows of the same length and direction represent the same vector.

Adding and subtracting vectors

To combine vectors, work one component at a time: add the $x$ components, the $y$ components, and (in three dimensions) the $z$ components.

Multiplying by a scalar

Multiplying a vector by a number stretches or shrinks it (and reverses it if the number is negative), without changing its line of direction. Every component is multiplied by the scalar.

A negative scalar reverses the direction: $-1$ times a vector points the opposite way with the same length.

The magnitude of a vector

The magnitude (or length) of a vector is found by Pythagoras, since the components form the sides of a right-angled triangle (or box in three dimensions).

In three dimensions the idea is identical, with a third squared term inside the root. The magnitude of $\begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}$ is $\sqrt{1^2 + 2^2 + 2^2} = \sqrt{9} = 3$.

The vector between two points

A vector can describe the journey from one point to another. The vector from $A$ to $B$, written $\overrightarrow{AB}$, is found by subtracting the position of $A$ from the position of $B$, component by component. This is the displacement from $A$ to $B$, and its magnitude is the straight-line distance between the points.

Examples in context

Vectors describe displacement, velocity and force, all of which have direction as well as size. A drone that flies $\begin{pmatrix} 30 \\ 40 \end{pmatrix}$ metres has travelled a straight-line distance of $\sqrt{30^2 + 40^2} = 50$ m, its magnitude. Adding a wind vector to an aircraft's velocity vector gives its true path over the ground. Engineers add force vectors nose to tail to find the single resultant force on a structure.

Try this

Q1. Find $\begin{pmatrix} 4 \\ 2 \end{pmatrix} + \begin{pmatrix} 1 \\ 6 \end{pmatrix}$. [1 mark]

• Cue. $\begin{pmatrix} 5 \\ 8 \end{pmatrix}$.

Q2. Find $2\begin{pmatrix} 3 \\ -2 \end{pmatrix}$. [1 mark]

• Cue. $\begin{pmatrix} 6 \\ -4 \end{pmatrix}$.

Q3. Find the magnitude of $\begin{pmatrix} 2 \\ 3 \\ 6 \end{pmatrix}$. [2 marks]

• Cue. $\sqrt{4 + 9 + 36} = \sqrt{49} = 7$.