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Exam: spins, distance and angle

Spin P(17, 7, 7) and line r (through (6, 5, 7), direction (2, 2, 4)) by 30° around the vertical axis through O(12, 3, 5). Then find the distance from P′ to plane α and the angle between r′ and α.

Step-by-step solution

1. Spin P around the axis: its height does not change and its distance to the axis is preserved.
2. Spin two points of r with the same axis and angle and join them to get r′.
3. Measure the distance from P′ to α with the perpendicular and foot F.
4. Find the angle between r′ and α using the normal.

Result: P′(14.33, 8.96, 7); r′ direction (0.73, 2.73, 4); P′F = 3.74 u; angle r′-α = 70.5°.

When spinning, the line does not change length and the point keeps its distance to the axis; use that as a check.

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