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Distance from a point to a line

Find the distance from point P(16, 18, 10) to line r, which passes through A(2, 7, 12) with direction (0, 0, 10).

Step-by-step solution

1. Draw through P the plane perpendicular to r.
2. Find point I where that plane cuts r: it is the foot of the perpendicular.
3. Segment PI is the distance, but only measurable in true length if it is in true length.
4. If the line is horizontal or vertical, the distance is seen directly in one of the views.

Result: I(2, 7, 10); distance PI = 17.8 units.

The foot of the perpendicular is not the closest point “by eye”: it is found by projecting P onto the direction of r.

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