Vectors and geometric proof

Vectors describe both magnitude (size) and direction. They appear in CCEA Higher tier and are often combined with geometric reasoning.

Vector notation

A vector can be written as:

• Bold letter: a or b.
• Column vector: $\begin{pmatrix} 3 \ -2 \end{pmatrix}$ (3 units right, 2 units down).
• Arrow notation: $\overrightarrow{AB}$ (from point A to point B).

Adding and subtracting vectors

Addition: place the second vector where the first ends (head-to-tail rule). a + b = travel along a then along b.

Subtraction: $\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA}$ = b − a (where O is the origin).

Negative vector: −a has the same magnitude as a but opposite direction.

Scalar multiplication

ka is a vector in the same direction as a with magnitude scaled by k. If k is negative, the direction reverses.

Magnitude of a vector

For $\begin{pmatrix} x \ y \end{pmatrix}$: magnitude = $\sqrt{x^2 + y^2}$ (Pythagoras).

Position vectors

If O is the origin, the position vector of point A is $\overrightarrow{OA}$ = a. $\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA}$ = b − a.

Vector paths and geometric proof

To find a vector path between two points, use the triangle law: $\overrightarrow{AC} = \overrightarrow{AB} + \overrightarrow{BC}$

Express everything in terms of the given vectors a and b (or p and q).

Proving collinearity: if $\overrightarrow{PQ} = k \overrightarrow{PR}$ (one is a scalar multiple of the other), and they share point P, then P, Q, R are collinear (on the same straight line).

Proving lines bisect: show that the midpoint vector of one line segment equals the midpoint vector of another.

CCEA examiner style

CCEA Higher vector questions typically:

1. Give position vectors or set up a diagram with vector labels.
2. Ask you to express $\overrightarrow{AB}$ in terms of given vectors.
3. Ask you to prove a geometric result (midpoint, collinearity, parallelism).

Always show all steps in the vector path — do not skip from given to answer.