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Line of maximum inclination

Find the line of maximum inclination of plane α (oblique plane, normal (1, 3, 3), constant 0) and its angle with the PV.

Step-by-step solution

Find the vertical trace α₂: the intersection of the plane with the PV.
The line of maximum inclination is perpendicular to α₂ and lies in the plane.
Its direction is the cross product of the plane normal and the direction of α₂.
The angle with the PV is measured between that line and its vertical projection, or between the line and the normal to the PV.
If the plane is parallel to the PV it has no vertical trace: the case is not defined.

Result: oblique plane. s passes through P(0, 0, 0) with direction (3, -10, 9); angle with the PV 46.51°.

Do not confuse maximum inclination with steepest slope: slope is measured against the PH (perpendicular to α₁) and inclination against the PV (perpendicular to α₂).

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