Connect lines, planes and angles

A line in space needs a point and a direction. A plane needs a point and a nonzero normal. A point lies on the plane when its displacement from the given point is perpendicular to the normal.

The cross product gives a vector perpendicular to two nonparallel vectors. Its magnitude is the parallelogram area. Substitute a line into a plane equation to test for an intersection.

\displaystyle \mathbf r=\mathbf a+t\mathbf d,\quad\mathbf n\cdot(\mathbf r-\mathbf p)=0

WORKED EXAMPLE

Find where r = (1, 0, 2) + t(1, 2, −1) meets x + y + z = 6.

Substitute: (1 + t) + 2t + (2 − t) = 6.
This gives 3 + 2t = 6, so t = $\frac{3}{2}$ .
The point is ( $\frac{5}{2}$ , 3, $\frac{1}{2}$ ).

Assumed knowledge

Calculate vector components, magnitudes and dot products, including the test for perpendicular vectors. Work with three-dimensional coordinates and solve linear equations by substitution.

Study Specialist Mathematics alongside, or after, Mathematical Methods. Later units build on earlier units.

QCAA prior-learning guidance ↗

Connect lines, planes and angles

Find where r = (1, 0, 2) + t(1, 2, −1) meets x + y + z = 6.

Put the idea to work.