Measure alignment with a dot product

The scalar product combines components to measure alignment. A positive value indicates an acute angle, a negative value an obtuse angle, and zero indicates perpendicular nonzero vectors.

A vector projection is the component of one vector along another. The dot product divided by the target magnitude gives a signed scalar projection; multiply by its unit direction for the vector projection.

\displaystyle \mathbf a\cdot\mathbf b=|\mathbf a||\mathbf b|\cos\theta,\quad \operatorname{proj}_{\mathbf b}\mathbf a=\frac{\mathbf a\cdot\mathbf b}{\mathbf b\cdot\mathbf b}\mathbf b

WORKED EXAMPLE

Are (2, 3) and (3, −2) perpendicular?

Their dot product is 2(3) + 3(−2) = 0.
Both vectors are nonzero, so the angle is 90°.

Assumed knowledge

Add and subtract plane vectors, multiply by a scalar, and calculate magnitudes and unit vectors. Use cosine and inverse cosine to relate an angle to side lengths.

Study Specialist Mathematics alongside, or after, Mathematical Methods. Later units build on earlier units.

QCAA prior-learning guidance ↗

Measure alignment with a dot product

Are (2, 3) and (3, −2) perpendicular?

Put the idea to work.