Question

a decorative light fixture is at coordinates (0, 0, 5). the fixture is rotated 90° counterclockwise about the positive x - axis.
what will its new coordinates be?
(-5, 0, 0)
(0, - 5, 0)
(5, 0, 0)
(0, 5, 0)

Explanation:

Step1: Recall rotation formula about x - axis

The rotation matrix about the \(x\) - axis for a point \((x,y,z)\) is \(

$$\begin{bmatrix}1&0&0\\0&\cos\theta&-\sin\theta\\0&\sin\theta&\cos\theta\end{bmatrix}$$

\). When \(\theta = 90^{\circ}\), \(\cos\theta=0\) and \(\sin\theta = 1\).

Step2: Apply rotation to the point \((0,0,5)\)

Let the original point be \(\vec{v}=

$$\begin{bmatrix}0\\0\\5\end{bmatrix}$$

\). After rotation, \(\vec{v}'=

$$\begin{bmatrix}1&0&0\\0&0&- 1\\0&1&0\end{bmatrix}$$

$$\begin{bmatrix}0\\0\\5\end{bmatrix}$$

=

$$\begin{bmatrix}0\\-5\\0\end{bmatrix}$$

\)

Answer:

\((0,-5,0)\)