Questions: Rotate the vector ⟨-3,5⟩ 270° clockwise about the origin. ⟨-5,[?]⟩

Transcript text: Rotate the vector $\langle-3,5\rangle 270^{\circ}$ clockwise about the origin. \[ \langle-5,[?]\rangle \]

Solution

Solution Steps

To rotate a vector $\langle x, y \rangle$ by $270^\circ$ clockwise about the origin, we can use the rotation matrix for $270^\circ$ clockwise, which is equivalent to $90^\circ$ counterclockwise. The rotation matrix for $90^\circ$ counterclockwise is: \[ \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \] We multiply this matrix by the vector $\langle x, y \rangle$ to get the new coordinates.

Step 1: Define the Original Vector

The original vector is given as $ \langle -3, 5 \rangle $.

Step 2: Apply the Rotation Matrix

To rotate the vector $ \langle x, y \rangle $ by $ 270^\circ $ clockwise, we use the rotation matrix for $ 90^\circ $ counterclockwise: \[ \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \] Multiplying this matrix by the vector $ \langle -3, 5 \rangle $: \[ \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} -3 \\ 5 \end{bmatrix}

\begin{bmatrix} 0 \cdot (-3) + (-1) \cdot 5 \\ 1 \cdot (-3) + 0 \cdot 5 \end{bmatrix}

\begin{bmatrix} -5 \\ -3 \end{bmatrix} \]

Step 3: Extract the New Coordinates

The new coordinates after the rotation are $ \langle -5, -3 \rangle $.