Example 8: Plane and Line Parallel?

Determine whether the plane 3x - 2y + z = 4 and the line X = (3i - j + k) + t(i + j - k) are parallel.

The plane has the normal vector

N = 3i - 2j + k.

The line has the direction vector

D = i + j - k.

We compute 

N · D = 3 · 1 + (- 2) · 1 + 1( - 1) = 0.

Therefore the plane and line are parallel (Figure 10.5.10).

Figure 10.5.10