The key to the equations of lines and planes in three dimensions is that, in each case, we need a point to locate the object in space, and a vector to tilt it at the correct angle. In each case, however, the kind of vector that unambiguously gives the direction of the object is different.

For a line, there is only one way to be parallel, but infinitely many ways to be orthogonal (think: every vector parallel to the xy-plane is orthogonal to the z-axis). Therefore the vector we want, the direction vector, is parallel to the line.

For a plane, conversely, there are infinitely many ways to be parallel, but only one way to be orthogonal (any vector orthogonal to the xy-plane is parallel to the z-axis). Therefore the vector we want, the normal vector, is orthogonal to the plane.

You may obtain these two pieces in many ways. In addition to being given the point and direction vector immediately, the following are enough to determine a line:

• two points on the line
• a point and a parallel line
• a point and two nonparallel vectors orthogonal to the line
• a point and two nonparallel lines orthogonal to the desired line
• a point and an orthogonal plane
• two intersecting (nonidentical) planes (the line of intersection)
• two intersecting (nonidentical) lines (the line through their point of intersection and orthogonal to both)

The following are enough to define a plane, in addition to being given a point and normal vector directly:

• three points in the plane
• a point and an orthogonal line
• a point and a line in the plane not containing that point
• two lines in the plane
• a point and a parallel plane
• a point and two planes orthogonal to the desired plane