Vector products, lines and planes

What you’ll learn

• Use cross and scalar triple products for geometry.
• Find plane equations, intersections and shortest distances.

Understand the idea

The cross product of two non-parallel vectors is perpendicular to both, with magnitude equal to the area of their parallelogram. The scalar triple product measures signed volume; its absolute value gives the volume of a parallelepiped and zero indicates coplanar vectors.

A plane is fixed by a point and a normal vector, or by a point and two independent directions. The cross product of two plane normals gives the direction of their line of intersection. Shortest distances are perpendicular distances: project a connecting vector onto a plane normal or onto the common normal of two skew lines.

Key formulae

$$|\mathbf a\times\mathbf b|=|\mathbf a||\mathbf b|\sin\theta$$

$$V=|\mathbf a\cdot(\mathbf b\times\mathbf c)|$$

$$\mathbf r\cdot\mathbf n=d,\qquad D_{\text{point,plane}}=\frac{|\mathbf p\cdot\mathbf n-d|}{|\mathbf n|}$$

$$D_{\text{skew}}=\frac{|(\mathbf b-\mathbf a)\cdot(\mathbf u\times\mathbf v)|}{|\mathbf u\times\mathbf v|}$$

Worked example

Find the perpendicular distance from $(1,2,3)$ to $2x-y+2z=3$.

1. The plane normal is $(2,-1,2)$ and has magnitude $3$.
2. Substitute the point into the left side minus $3$: $2-2+6-3=3$.
3. Divide the absolute residual by the normal's magnitude.

Answer: $1$ unit.