Geometric significance of the cross product

Parametric equations

For $\mathbf{a},\mathbf{b}\in {\mathbb{R}}^3$, the cross product $\mathbf{a}\times \mathbf{b}$ gives both an area and a normal:

Here $\theta$ is the angle between the two vectors, from $0^{\circ }$ to $180^{\circ }$, so $\sin \theta$ is not negative. The magnitude is the area of the parallelogram with sides $\mathbf{a}$ and $\mathbf{b}$. The vector $\mathbf{a}\times \mathbf{b}$ is perpendicular to the plane they span, and the right-hand rule fixes the direction: fingers curl from $\mathbf{a}$ toward $\mathbf{b}$, and the thumb points along $\mathbf{n}$. If $\mathbf{a}$ and $\mathbf{b}$ are parallel, or either one is the zero vector, then $\mathbf{a}\times \mathbf{b}=0$.

On this figure $|\mathbf{a}|$ is fixed at $3$. The length $|\mathbf{b}|$ runs from $0.5$ to $4$, so neither vector is zero. The product is zero only at $\theta =0^{\circ }$ and $\theta =180^{\circ }$, where the parallelogram collapses to a line. The tilt $\phi$ runs from $-90^{\circ }$ to $90^{\circ }$ and rotates $\mathbf{b}$ around $\mathbf{a}$. The plane tilts with it. The angle and the area do not.

Interaction

• Angle θ: drag $\theta$ and watch the area follow $\Vert \mathbf{a}\times \mathbf{b}\Vert =|\mathbf{a}||\mathbf{b}|\sin \theta$. Near $0^{\circ }$ or $180^{\circ }$ the parallelogram flattens to a line
• Length |b|: change only $|\mathbf{b}|$. The area scales with it, and the direction of the normal stays the same
• Tilt φ: rotate $\mathbf{b}$ around $\mathbf{a}$. The plane and the normal turn together. The angle and the area stay put
• Area ‖a × b‖ / Right-hand rule: switch the reading between the area and the direction
• Drag the figure to rotate the view and check that $\mathbf{n}$ is perpendicular to the plane of $\mathbf{a}$ and $\mathbf{b}$. Horizontal view and Elevation are the same rotation

What to notice

• The cross product measures the area of the parallelogram the two vectors span. That area is $\Vert \mathbf{a}\times \mathbf{b}\Vert$.
• The direction of $\mathbf{a}\times \mathbf{b}$ is perpendicular to the plane of $\mathbf{a}$ and $\mathbf{b}$, and the right-hand rule fixes it. The figure draws $\mathbf{a}\times \mathbf{b}$.
• Swapping the order reverses the normal: $\mathbf{b}\times \mathbf{a}=-\mathbf{a}\times \mathbf{b}$.
• When the vectors are parallel, or one of them is the zero vector, the cross product is zero and they do not span a plane. This figure cannot make a vector zero, so the product vanishes only in the parallel case.

Related

• Spatial vector and three plane projections
• Plane normal and distance from a point
• Vector projection
• Spatial vectors, planes, and lines

Further reading

• Cross product (Wikipedia)
• Right-hand rule (Wikipedia)