Question

justine reflects a vector v across the y - axis. simonne reflects a vector v by rotating it 90° counterclockwise about the origin. which of the following vectors has the same image for justine and simonne?
\\(\

$$\begin{bmatrix}4\\\\4\\end{bmatrix}$$

\\)
\\(\

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

\\)
\\(\

$$\begin{bmatrix}4\\\\3\\end{bmatrix}$$

\\)
\\(\

$$\begin{bmatrix}-3\\\\3\\end{bmatrix}$$

\\)

Explanation:

Step1: Find the reflection across the \(y -\)axis

The transformation matrix for reflection across the \(y -\)axis is \(R_y=

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

\). If a vector \(\vec{v}=

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

\), then \(R_y\vec{v}=

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

\)

Step2: Find the rotation of \(90^{\circ}\) counter - clockwise about the origin

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

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

\). If a vector \(\vec{v}=

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

\), then \(R_{90}\vec{v}=

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

\)

Step3: Check each vector

• For \(\vec{v}=

$$\begin{bmatrix}4\\4\end{bmatrix}$$

\):

• Reflection across \(y -\)axis: \(R_y

$$\begin{bmatrix}4\\4\end{bmatrix}$$

=

$$\begin{bmatrix}- 4\\4\end{bmatrix}$$

\)

• Rotation of \(90^{\circ}\) counter - clockwise: \(R_{90}

$$\begin{bmatrix}4\\4\end{bmatrix}$$

=

$$\begin{bmatrix}-4\\4\end{bmatrix}$$

\)

• For \(\vec{v}=

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

\):

• Reflection across \(y -\)axis: \(R_y

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

=

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

\)

• Rotation of \(90^{\circ}\) counter - clockwise: \(R_{90}

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

=

$$\begin{bmatrix}1\\1\end{bmatrix}$$

\)

• For \(\vec{v}=

$$\begin{bmatrix}4\\3\end{bmatrix}$$

\):

• Reflection across \(y -\)axis: \(R_y

$$\begin{bmatrix}4\\3\end{bmatrix}$$

=

$$\begin{bmatrix}-4\\3\end{bmatrix}$$

\)

• Rotation of \(90^{\circ}\) counter - clockwise: \(R_{90}

$$\begin{bmatrix}4\\3\end{bmatrix}$$

=

$$\begin{bmatrix}-3\\4\end{bmatrix}$$

\)

• For \(\vec{v}=

$$\begin{bmatrix}-3\\3\end{bmatrix}$$

\):

• Reflection across \(y -\)axis: \(R_y

$$\begin{bmatrix}-3\\3\end{bmatrix}$$

=

$$\begin{bmatrix}3\\3\end{bmatrix}$$

\)

• Rotation of \(90^{\circ}\) counter - clockwise: \(R_{90}

$$\begin{bmatrix}-3\\3\end{bmatrix}$$

=

$$\begin{bmatrix}-3\\-3\end{bmatrix}$$

\)

Answer:

\(

$$\begin{bmatrix}4\\4\end{bmatrix}$$

\)