The cross product

The dot product answered "how much do these agree?" with a number. In 3D there is a different question: what direction is perpendicular to both of these? Its answer is a vector.

Drag to orbit. Two vectors span a parallelogram; the amber arrow stands perpendicular to that whole sheet.

Computing one

For $a = (a_1,a_2,a_3)$ and $b = (b_1,b_2,b_3)$:

$$a \times b = (a_2b_3 - a_3b_2,;; a_3b_1 - a_1b_3,;; a_1b_2 - a_2b_1)$$

Try $\hat{\imath} \times \hat{\jmath}$, that is $(1,0,0) \times (0,1,0)$:

• first entry: $0(0) - 0(1) = 0$
• second: $0(0) - 1(0) = 0$
• third: $1(1) - 0(0) = 1$

The result is $(0,0,1) = \hat{k}$. Two vectors lying flat in the floor produce one pointing straight up. Perpendicular to both, exactly as advertised.

The length is not decoration

$|a \times b|$ equals the area of the parallelogram that $a$ and $b$ span.

That connects it straight back to Act III. The determinant measured how area and volume change; here area shows up again, now attached to a direction. If you build a $3 \times 3$ matrix whose rows are $a$, $b$, and a third vector $c$, its determinant is exactly $(a \times b) \cdot c$ — the volume of the box they span. The two ideas were the same machinery all along.

Slide $b$ until it lines up with $a$ in the visual. The parallelogram flattens, its area goes to zero, and the cross product vanishes. Parallel vectors span no area, so there is no perpendicular direction to name.

Orientation, again

Two directions are perpendicular to any sheet: up and down. Which one you get is fixed by the right-hand rule, and $a \times b = -(b \times a)$. Order matters, and swapping it flips the result — the same orientation bookkeeping that made determinants go negative.

Now back to the main thread. You have tools for measuring; time to use them for the question this whole path has been walking toward.