Question

rotate the given triangle 90° counterclockwise about the origin.
\\( \left \

$$\begin{array} { c c c } { - 1 } & { 2 } & { 2 } \\\\ { - 1 } & { - 1 } & { 3 } \\end{array}$$

\
ight \\)
\\( \left \

$$\begin{array} { c c c } { 1 } & {? } & { } \\\\ { - 1 } & { } & { } \\end{array}$$

\
ight \\)

Explanation:

Step1: Recall rotation formula

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

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

\). If a point \((x,y)\) is rotated \(90^{\circ}\) counter - clockwise about the origin, the new point \((x',y')\) is given by \(

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

=

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

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

=

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

\).

Step2: Apply rotation to the second point

The second column of the original matrix represents the point \((2,-1)\).
Using the formula \(

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

=

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

\), when \(x = 2\) and \(y=-1\), we have \(x'=-(-1)=1\) and \(y' = 2\).

Answer:

The value in the green box (the first element of the second column of the rotated matrix) is \(1\), and the second element of the second column is \(2\).