Question

The Cartesian coordinates of a point P in a right-handed coordinate system are (1, 1, 1). The transformed coordinates of P due to a 45° clockwise rotation of the coordinate system about the positive x-axis are

• A. \( (1, 0, \sqrt{2}) \)
• B. \( (1, 0, -\sqrt{2}) \)
• C. \( (-1, 0, \sqrt{2}) \)
• D. \( (-1, 0, -\sqrt{2}) \)

Hint:

When rotating a point about the x-axis, the x-coordinate remains unchanged, while the y and z coordinates are transformed based on the rotation matrix.

Answer 1

The Correct Option is A

We are given that the point \( P \) has Cartesian coordinates \( (1, 1, 1) \). The transformation involves a 45° clockwise rotation about the positive x-axis. This rotation affects the y and z coordinates while leaving the x-coordinate unchanged. The rotation matrix for a 45° clockwise rotation about the x-axis is: \[ R = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos 45^\circ & -\sin 45^\circ \\ 0 & \sin 45^\circ & \cos 45^\circ \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ 0 & \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{pmatrix} \] Now, applying this rotation to the point \( P(1, 1, 1) \), we get the transformed coordinates: \[ \begin{pmatrix} x' \\ y' \\ z' \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ 0 & \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{pmatrix} \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \\ \sqrt{2} \end{pmatrix} \] Thus, the transformed coordinates are \( (1, 0, \sqrt{2}) \). Final Answer: (A) \( (1, 0, \sqrt{2}) \)