The Scalar Product — Pure Mathematics 3 (9709)

Core Concept

Given two vectors $\mathbf{a}$ and $\mathbf{b}$, the scalar product measures "how much" one vector points in the direction of the other. Geometrically, if $\theta$ is the angle between the vectors when placed tail-to-tail ($0° \leq \theta \leq 180°$), then:

Key geometric consequences:

• If $\mathbf{a} \cdot \mathbf{b} = 0$ and neither vector is zero, then $\cos\theta = 0$, so $\theta = 90°$: the vectors are perpendicular.
• If $\mathbf{a} \cdot \mathbf{b} > 0$ then $\theta$ is acute; if $\mathbf{a} \cdot \mathbf{b} < 0$ then $\theta$ is obtuse.

Angle between two lines: Lines have directions, not orientations — so the angle between two lines is always taken as the acute (or right) angle, i.e. between $0°$ and $90°$. If the formula gives an obtuse angle $\theta$, use $180° - \theta$ instead.

Foot of the perpendicular: Given a point $A$ not on line $\ell$, the foot of the perpendicular $F$ is the unique point on $\ell$ such that $\overrightarrow{AF} \perp \ell$. This is found by expressing $F$ in terms of the parameter $\lambda$, then imposing $\overrightarrow{AF} \cdot \mathbf{d} = 0$ (where $\mathbf{d}$ is the direction vector of $\ell$), and solving for $\lambda$.