Questions: Choose the correct vector as a result of applying the rotation matrix to v, shown in the graph to the right. [2-sqrt(3), 1+sqrt(3)] [-4, -2] [-2, 4] Rotation Matrix [-1 0, 0 -1]

Choose the correct vector as a result of applying the rotation matrix to v, shown in the graph to the right.
[2-sqrt(3), 1+sqrt(3)]
[-4, -2]
[-2, 4]
Rotation Matrix [-1 0, 0 -1]

Transcript text: Choose the correct vector as a result of applying the rotation matrix to $v$, shown in the graph to the right. $\left[\begin{array}{l}2-\sqrt{3} \\ 1+\sqrt{3}\end{array}\right]$ $\left[\begin{array}{r}-4 \\ -2\end{array}\right]$ $\left[\begin{array}{c}-2 \\ 4\end{array}\right]$ Rotation Matrix $\left[\begin{array}{cc}-1 & 0 \\ 0 & -1\end{array}\right]$ DONE

Solution

Solution Steps

Step 1: Identify the given vector and rotation matrix

The given vector $ v $ is $\begin{bmatrix} 4 \\ 2 \end{bmatrix}$ and the rotation matrix is $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$.

Step 2: Set up the matrix multiplication

To find the rotated vector, we need to multiply the rotation matrix by the vector $ v $: \[ \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 4 \\ 2 \end{bmatrix} \]

Step 3: Perform the matrix multiplication

Calculate the resulting vector by performing the matrix multiplication: \[ \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 4 \\ 2 \end{bmatrix} = \begin{bmatrix} (0 \cdot 4) + (-1 \cdot 2) \\ (1 \cdot 4) + (0 \cdot 2) \end{bmatrix} = \begin{bmatrix} -2 \\ 4 \end{bmatrix} \]