Line of steepest slope

Find the line of steepest slope of plane α (normal (2, 1, 2), constant -58) and its angle with the PH.

Step-by-step solution

Find the horizontal trace α₁: the intersection of the plane with the PH.
The line of steepest slope is perpendicular to α₁ and lies in the plane.
Its direction is the cross product of the plane normal and the direction of α₁.
The angle with the PH is measured by rotating flat the projecting plane that contains that line.

Result: s passes through P(18, 12, 5) with direction (4, 2, -5); slope 111.8 % (48.19°).

The steepest slope is the line rotated flat to measure the inclination of the plane; do not confuse it with any line of the plane.