11b.6. Geometrical meanings

Let us recap the key ideas and formulas in the chapter:

$$ \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| \, |\mathbf{b}| \cos \theta \\ |\mathbf{a} \times \mathbf{b}| = |\mathbf{a}| \, |\mathbf{b}| \sin \theta \\ \textrm{Area of parallelogram formed by } \mathbf{a} \textrm{ and } \mathbf{b} = |\mathbf{a} \times \mathbf{b}| \\ \textrm{Area of triangle } OAB = \frac{1}{2}|\mathbf{a} \times \mathbf{b}| $$

If $\mathbf{a}$ is perpendicular to $\mathbf{b}$, then $\mathbf{a} \cdot \mathbf{b} = 0$. Conversely, if $\mathbf{a} \cdot \mathbf{b} = 0$, then either $\mathbf{a} = \mathbf{0}, \mathbf{b} = \mathbf{0}$ or $\mathbf{a}$ is perpendicular to $\mathbf{b}$.
The latter case answers example 3.

Similarly, if $\mathbf{a}$ is parallel to $\mathbf{b}$, then $\mathbf{a} \times \mathbf{b} = \mathbf{0}$. Conversely, if $\mathbf{a} \times \mathbf{b} = \mathbf{0}$, then either $\mathbf{a} = \mathbf{0}, \mathbf{b} = \mathbf{0}$ or $\mathbf{a}$ is parallel to $\mathbf{b}$.
The latter case answers example 4.

We also have the picture and formulas for projection and perpendicular lengths:

These formulas and discussions above answer the following example questions about the interpretation of certain formulas and equations:

Solution to examples

1. Give a geometrical interpretation of $|\mathbf{a} \cdot \hat{\mathbf{b}}$.

It is the length of projection of $\mathbf{a}$ onto $\mathbf{b}$.

2. If $\mathbf{c}$ is a unit vector, give a geometrical interpretation of $|\mathbf{a} \times \mathbf{c}|$.

Since $\mathbf{c}$ is a unit vector, $\mathbf{c} = \hat{\mathbf{c}}$. Hence $|\mathbf{a} \times \mathbf{c}|$ is the perpendicular length from point $A$ to the line $OC$.