Compute the scalar product of two vectors in component or geometric form and use it to find the angle between vectors and test orthogonality

What this dot point is asking

NESA wants you to compute the scalar (dot) product of two vectors using either the component formula or the geometric formula, recognise its meaning as a measure of alignment, and use it to find the angle between vectors and to test orthogonality.

The answer

Two equivalent formulas

For two-dimensional vectors $\mathbf{a} = (a_1, a_2)$ and $\mathbf{b} = (b_1, b_2)$, the scalar product (dot product) is

Geometrically,

where $\theta$ is the angle between the vectors (with $0 \le \theta \le \pi$).

The two formulas are equivalent: they describe the same number.

The angle between two vectors

Solving the geometric formula for $\cos \theta$,

This works for any two non-zero vectors. To find $\theta$, apply $\arccos$ (the principal value).

Orthogonality

Two non-zero vectors are perpendicular (orthogonal) if and only if $\mathbf{a} \cdot \mathbf{b} = 0$.

This is the standard test: instead of computing the angle and checking it equals $90^\circ$, just set the dot product to zero.

Properties

The scalar product is commutative ($\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}$), bilinear (linear in each argument), and gives $\mathbf{a} \cdot \mathbf{a} = |\mathbf{a}|^2$.

These let you manipulate dot products like algebraic products (with the caveat that the result is a scalar, not a vector).

The geometric meaning

$\mathbf{a} \cdot \mathbf{b}$ measures the extent to which $\mathbf{a}$ and $\mathbf{b}$ point in the same direction.

• IMATH_21 : the angle between them is acute ($< 90^\circ$).
• IMATH_23 : they are perpendicular.
• IMATH_24 : the angle is obtuse ($> 90^\circ$).

Use in determining work and projection

The scalar product appears in physics as work $W = \mathbf{F} \cdot \mathbf{d}$, and in projection (next dot point). For pure mathematics, the angle calculation is the most common application.