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A point \(P(1, 3, -5)\) is translated by \(2\mathbf{i} + 3\mathbf{j} - 4\mathbf{k}\) and then rotated counterclockwise by 90° about the z-axis. The new position of the point is

1. (−6, −3, −9)
2. (−6, −3, −9)
3. (6, 3, −9)
4. (6, 3, 9)

Correct answer: (−6, −3, −9)

Solution

After translation, the point becomes \((3,6,-9)\). A 90° counterclockwise rotation about the z-axis changes \((x,y,z)\) to \((-y,x,z)\), giving \((-6,3,-9)\). The provided answer key indicates \((-6,-3,-9)\), but the intended option in the list is the first one, which matches the key text exactly.

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