Question

rotate the vector $langle 0,2
angle$ $270^{circ}$ degrees counterclockwise about the origin. $langle square,square
angle$

Explanation:

Step1: Recall the rotation matrix

The rotation matrix for a counter - clockwise rotation by an angle \(\theta\) is \(R(\theta)=

$$\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}$$

\). For \(\theta = 270^{\circ}\), \(\cos(270^{\circ}) = 0\) and \(\sin(270^{\circ})=- 1\). So \(R(270^{\circ})=

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

\).

Step2: Apply the rotation matrix to the vector

The vector \(\vec{v}=

$$\begin{bmatrix}0\\2\end{bmatrix}$$

\). Multiply the matrix \(R(270^{\circ})\) and \(\vec{v}\):

$$
LATEXBLOCK3
$$

Answer:

\(\langle2,0
angle\)