Question

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

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

\
ight$

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 each point

Let the original matrix be \(M=

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

\).
For a point \((x,y)\) in the original matrix, after rotation, it becomes \((y,-x)\).
For the point \((-3,1)\):
After rotation, \(x=-3,y = 1\), the new point is \((1,3)\).
For the point \((5,2)\):
After rotation, \(x = 5,y=2\), the new point is \((2,-5)\).

Answer:

The completed matrix is \(

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

\). So the green box (first - row, third - column) is \(2\), the first gray box (second - row, second - column) is \(3\), and the second gray box (second - row, third - column) is \(-5\).