Angle between a line and a plane

Find the angle between line r (through P(8, 4, 11), direction (3, 5, 7)) and plane α with normal (1, 3, 3) through A(13, 6, 7).

Step-by-step solution

The angle between a line and a plane is the one it makes with its own projection on the plane.
If the plane is projecting, the line and its projection are seen in true size in the edge-on view.
If not, rotate flat the plane formed by the line and its projection, or use the complement of the angle with the normal.
Check: if the line lies in the plane, 0°; if it is perpendicular, 90°.

Result: 79.14°.

The angle with the plane and the angle with the normal add up to 90°. It is the shortcut that saves the most time in an exam.