Question

rotate the given triangle $90^{circ}$ clockwise about the origin.
$\

$$\begin{bmatrix}0&-3&5\\\\0&1&2\\end{bmatrix}$$

$
$\

$$\begin{bmatrix}0&?&\\\\0&&\\end{bmatrix}$$

$

Explanation:

Step1: Recall the rotation matrix

The rotation matrix for a \(90^{\circ}\) clockwise rotation about the origin is \(

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

\).

Step2: Apply the rotation to the point \((-3,1)\)

Let the point be \((x,y)=(-3,1)\). Using the formula \(

$$\begin{bmatrix}x'\\y'\end{bmatrix}$$

=

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

$$\begin{bmatrix}x\\y\end{bmatrix}$$

\), we get \(x' = 0\times(-3)+1\times1 = 1\) and \(y'=-1\times(-3)+0\times1 = 3\).

Answer:

\(1\)