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Exam: section, penetration, distance and angle

The cube of centre (10, 8, 3.75) is cut by plane α (normal (2, 2, 2), constant -44.06). Find: 1) the section and its area; 2) the entry and exit points of line r (through (10, 8, 3.75), direction (2, 5, 2)); 3) the distance from P(11, 13, 9) to α; 4) the angle between r and α.

Step-by-step solution

1. Find the section of the cube by α and its true shape (area and perimeter).
2. Put r inside a projecting plane and find its entry and exit points in the cube.
3. Draw through P the perpendicular to α, find the foot F and measure PF.
4. The angle between r and α is the complement of the angle between r and the normal.
5. Check every result with the plane equation and the cube mesh.

Result: section with 6 vertices, area 72.94 u²; it enters and leaves the cube; PF = 6.33 u; angle r-α = 64.76°.

It is a chaining exercise: the section does not affect the distance, but the order of the steps helps you not to leave anything half done.

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