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A rectangle has vertices at P1=(5,3,-3), P2=(5,9,9), P3=(0,5,11) and P4=(0,-1,-1). It is rotated about its own diagonal (with direction cosines (1/3, 2/3, 2/3)) until the new position of the rectangle is perpendicular to its original position. Which of the following could be a vertex of the rectangle in its new position?

1. one of the vertices in new position can be (5,-3,3)
2. one of the vertices in new position can be (0,11,5)
3. one of the vertices in new position can be (-3,5,-1)
4. one of the vertices in new position can be (8,3,9)

Correct answer: one of the vertices in new position can be (5,-3,3)

Solution

The rectangle rotates 90 deg about its diagonal. Vertices on the diagonal remain fixed. The other two vertices rotate about the diagonal axis. Applying Rodrigues' formula for 90 deg rotation about n_hat = (1/3, 2/3, 2/3) to the non-diagonal vertices, one checks which of the given points matches the rotated positions.

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