Dot product and cross product

What is it about?

The dot product $\vec a \cdot \vec b = a_1b_1 + a_2b_2 + a_3b_3$ is a number. Since $\vec a \cdot \vec b = |\vec a|\,|\vec b| \cos\varphi$, it gives the angle: $(1, 0, 0)^T \cdot (1, 1, 0)^T = 1$, $\cos\varphi = \frac{1}{\sqrt2}$, $\varphi = 45°$. $\vec a \cdot \vec b = 0$ means perpendicular; $\frac{\vec a \cdot \vec b}{|\vec a|}$ is the signed length of the projection of $\vec b$ onto $\vec a$.

The cross product $\vec a \times \vec b = (a_2b_3 - a_3b_2,\ a_3b_1 - a_1b_3,\ a_1b_2 - a_2b_1)^T$ is perpendicular to $\vec a$ and $\vec b$ (right-hand rule), e.g. $(1, 0, 0)^T \times (0, 1, 0)^T = (0, 0, 1)^T$. Its length $|\vec a|\,|\vec b| \sin\varphi$ is the area of the parallelogram. For parallel vectors it is $\vec o$.

The triple product $(\vec a \times \vec b) \cdot \vec c$ gives the volume (base times height); shearing keeps it (Cavalieri). A negative sign means a left-handed system.

Try it

1. Turn $\varphi$ from $0°$ to $180°$ and compare both curves. Rotate
2. Watch the cross product step by step. Cross product
3. Shear the parallelepiped upright. Cavalieri
4. Find three perpendicular vectors. Discover

Tasks

Task 1: Angle

Find the angle between $(1, 2, 0)^T$ and $(-2, -1, 2)^T$. Load task

Task 2: Area

Compute $(1, 2, 3)^T \times (4, 5, 6)^T$ and the area of the parallelogram. Load task

Task 3: All solutions

Which $\vec c$ give volume $12$ with $\vec a = (2, 0, 0)^T$, $\vec b = (0, 3, 0)^T$? Load task

Ideas for the classroom

• Searching for perpendicular vectors in “Discover” first leads naturally to the normal form of a plane.
• Common stumbling blocks: writing the dot product as a vector; sign errors in the middle component (check with $(\vec a \times \vec b) \cdot \vec a = 0$); forgetting $\frac12$ or $\frac16$.