Unit 7
Lines and Planes in Three Dimensions

In this unit, we use vector algebra as a tool to study properties of lines and planes in three-dimensional space. If you recall the discussion on visualizing linear equations from Unit 1, you know that a single linear equation in three unknowns is the equation of a plane in three dimensions. You will find it useful to reread this discussion before proceeding with this unit.

In this unit, you will learn the definition of a “normal” vector, the “point-normal” and “general” forms for the equation of a plane, and the “parametric” and “symmetric” forms for the equation of a straight line.

Lines and planes in three dimensions are example of linear subspaces, a topic you will study in Units 8 and 9.

Objectives

When you have completed this unit, you should be able to

1. find the point-normal and general forms of the equation of a plane passing through three noncollinear points.
2. find, given a vector or straight line and a point, the equation of a plane through that point and orthogonal to the given vector or straight line.
3. find the parametric, vector, and symmetric forms for the equations of a straight line.
4. find the distance from a point to a plane, and the distance between two parallel planes.