Vectors in two and three dimensions, magnitude and direction, addition and scalar multiplication, unit vectors and components, position vectors, and using vectors to solve geometric problems.

What this dot point is asking

AQA wants you to use vectors in two and three dimensions, find magnitude and direction, add vectors and multiply by scalars, work with unit vectors and components, use position vectors, and apply vectors to solve geometric problems such as proving points are collinear or finding a midpoint. The geometric applications, collinearity, parallelism and distances, are where most exam marks lie.

Components and notation

A vector in three dimensions can be written as $\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k}$ or as a column vector. The unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ point along the three coordinate axes and each have magnitude one. Two vectors are equal only when all their corresponding components match, which means equal magnitude and equal direction together.

Magnitude and unit vectors

Addition, scaling and position vectors

Vectors add component by component, and multiplying by a scalar stretches the vector (and reverses it if the scalar is negative). The position vector of a point $A$ is $\overrightarrow{OA}$, measured from the origin. The displacement from $A$ to $B$ is $\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA}$, the key relationship for almost every vector geometry question.

Geometric applications

Two vectors are parallel if one is a scalar multiple of the other, written $\mathbf{u} = k\mathbf{v}$. Three points are collinear if the vector between one pair is a scalar multiple of the vector between another pair that shares a common point: showing $\overrightarrow{AB} = k\overrightarrow{BC}$ with $B$ shared proves $A$, $B$, $C$ lie on a line. The midpoint of $A$ and $B$ has position vector $\frac{1}{2}(\overrightarrow{OA} + \overrightarrow{OB})$. To divide $AB$ in a given ratio, add the appropriate fraction of $\overrightarrow{AB}$ to $\overrightarrow{OA}$.

Direction, angles and a general method

The direction of a two-dimensional vector $a_1\mathbf{i} + a_2\mathbf{j}$ is given by the angle it makes with a chosen axis, found from $\tan\theta = \frac{a_2}{a_1}$ (taking care with the quadrant). In three dimensions, you typically describe direction through the unit vector rather than a single angle. A vector and its negative point in opposite directions but have the same magnitude, and scaling by a positive factor preserves direction while a negative factor reverses it.

Most vector geometry questions yield to one general method: convert every point to a position vector, express the displacements you need as differences of position vectors, and then translate the geometric condition into algebra. Parallel becomes "one displacement is a scalar multiple of another"; collinear becomes "parallel displacements sharing a point"; a midpoint becomes the average of two position vectors; and a distance becomes the magnitude of a displacement. Setting the problem up in this consistent way turns wordy geometry into routine component arithmetic, which is where the marks are reliably earned.