To rotate the vector [tex]\(\langle -3, 5 \rangle\)[/tex] by [tex]\(270^\circ\)[/tex] clockwise about the origin, we need to apply a rotation matrix appropriate for a clockwise rotation.
1. Angle Conversion:
The angle of rotation given is [tex]\(270^\circ\)[/tex].
2. Rotation Matrix:
The general form of the rotation matrix for a clockwise rotation by angle [tex]\(\theta\)[/tex] is:
[tex]\[
\begin{bmatrix}
\cos(\theta) & \sin(\theta) \\
-\sin(\theta) & \cos(\theta)
\end{bmatrix}
\][/tex]
For [tex]\(\theta = 270^\circ\)[/tex], we need to use the trigonometric values for [tex]\(270^\circ\)[/tex].
3. Trigonometric Values:
We know that:
[tex]\[
\cos(270^\circ) = 0 \quad \text{and} \quad \sin(270^\circ) = -1
\][/tex]
4. Substitute Trigonometric Values:
Substituting these values into the rotation matrix, we get:
[tex]\[
\begin{bmatrix}
0 & -1 \\
1 & 0
\end{bmatrix}
\][/tex]
5. Matrix Multiplication:
Now, we multiply this rotation matrix by our vector [tex]\(\langle -3, 5 \rangle\)[/tex]:
[tex]\[
\begin{bmatrix}
0 & -1 \\
1 & 0
\end{bmatrix}
\begin{bmatrix}
-3 \\
5
\end{bmatrix}
\][/tex]
Calculating the resulting vector:
[tex]\[
\begin{bmatrix}
0 \cdot -3 + (-1) \cdot 5 \\
1 \cdot -3 + 0 \cdot 5
\end{bmatrix}
=
\begin{bmatrix}
-5 \\
-3
\end{bmatrix}
\][/tex]
So, the vector [tex]\(\langle -3, 5 \rangle\)[/tex] rotated [tex]\(270^\circ\)[/tex] clockwise about the origin is [tex]\(\langle -5, -3 \rangle\)[/tex].
Therefore, the rotated vector [tex]\(\langle -3, 5 \rangle\)[/tex] is [tex]\(\langle -4.999999999999999, -3.000000000000001 \rangle\)[/tex].
