https://r2.erweima.ai/mathbot/image/1722364748417ft1w9327.png

To solve this problem, we need to rotate the given vector \(\begin{bmatrix} -3 \\ -5 \end{bmatrix}\) by various angles and match the resulting vector to the correct angle. The rotation of a vector by an angle \(\theta\) is given by the matrix multiplication:

\[
\begin{bmatrix}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{bmatrix}
\begin{bmatrix}
-3 \\
-5
\end{bmatrix}
\]

We'll calculate the rotated vectors for each provided angle and match them to the given results.

### Step-by-Step Calculations

1. **Angle \(\theta = \frac{\pi}{4}\)**

\[
\cos\left(\frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}
\]

\[
\begin{bmatrix}
\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\
\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}
\end{bmatrix}
\begin{bmatrix}
-3 \\
-5
\end{bmatrix}
=
\begin{bmatrix}
\frac{\sqrt{2}}{2}(-3) + -\frac{\sqrt{2}}{2}(-5) \\
\frac{\sqrt{2}}{2}(-3) + \frac{\sqrt{2}}{2}(-5)
\end{bmatrix}
=
\begin{bmatrix}
1.41 \\
-5.66
\end{bmatrix}
\]

2. **Angle \(\theta = \frac{\pi}{2}\)**

\[
\cos\left(\frac{\pi}{2}\right) = 0, \quad \sin\left(\frac{\pi}{2}\right) = 1
\]

\[
\begin{bmatrix}
0 & -1 \\
1 & 0
\end{bmatrix}
\begin{bmatrix}
-3 \\
-5
\end{bmatrix}
=
\begin{bmatrix}
5 \\
-3
\end{bmatrix}
\]

3. **Angle \(\theta = \frac{2\pi}{3}\)**

\[
\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}, \quad \sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2}
\]

\[
\begin{bmatrix}
-\frac{1}{2} & -\frac{\sqrt{3}}{2} \\
\frac{\sqrt{3}}{2} & -\frac{1}{2}
\end{bmatrix}
\begin{bmatrix}
-3 \\
-5
\end{bmatrix}
=
\begin{bmatrix}
0.1 \\
5.83
\end{bmatrix}
\]

4. **Angle \(\theta = \frac{4\pi}{3}\)**

\[
\cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2}, \quad \sin\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{2}
\]

\[
\begin{bmatrix}
-\frac{1}{2} & \frac{\sqrt{3}}{2} \\
-\frac{\sqrt{3}}{2} & -\frac{1}{2}
\end{bmatrix}
\begin{bmatrix}
-3 \\
-5
\end{bmatrix}
=
\begin{bmatrix}
-5.66 \\
-1.41
\end{bmatrix}
\]

5. **Angle \(\theta = \frac{5\pi}{3}\)**

\[
\cos\left(\frac{5\pi}{3}\right) = \frac{1}{2}, \quad \sin\left(\frac{5\pi}{3}\right) = -\frac{\sqrt{3}}{2}
\]

\[
\begin{bmatrix}
\frac{1}{2} & \frac{\sqrt{3}}{2} \\
-\frac{\sqrt{3}}{2} & \frac{1}{2}
\end{bmatrix}
\begin{bmatrix}
-3 \\
-5
\end{bmatrix}
=
\begin{bmatrix}
-5.66 \\
-1.41
\end{bmatrix}
\]

6. **Angle \(\theta = \frac{7\pi}{4}\)**

\[
\cos\left(\frac{7\pi}{4}\right) = \sin\left(\frac{7\pi}{4}\right) = \frac{\sqrt{2}}{2}
\]

\[
\begin{bmatrix}
\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\
-\\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}
\end{bmatrix}
\begin{bmatrix}
-3 \\
-5
\end{bmatrix}
=
\begin{bmatrix}
1.41 \\
-5.66
\end{bmatrix}
\]

### Matching Results

- \(\begin{bmatrix} 5.83 \\ -0.10 \end{bmatrix}\) matches \(\frac{2\pi}{3}\)
- \(\begin{bmatrix} -5.66 \\ -1.41 \end{bmatrix}\) matches \(\frac{4\pi}{3}\)
- \(\begin{bmatrix} -2.83 \\ 5.10 \end{bmatrix}\) matches \(\frac{\pi}{2}\)
- \(\begin{bmatrix} 5 \\ -3 \end{bmatrix}\) matches \(\frac{\pi}{2}\)

### Final Pairings
\[
\begin{bmatrix} 5.83 \\ -0.10 \end{bmatrix} \rightarrow \frac{2\pi}{3}
\]
\[
\begin{bmatrix} -5.66 \\ -1.41 \end{bmatrix} \rightarrow \frac{4\pi}{3}
\]
\[
\begin{bmatrix} -2.83 \\ 5.10 \end{bmatrix} \rightarrow \frac{\pi}{2}
\]
\[
\begin{bmatrix} 5 \\ -3 \end{bmatrix} \rightarrow \frac{\pi}{2}
\]

Do you have any questions or need further details?

Here are some related questions:

1. What is the formula for rotating a vector by a given angle?
2. How do you derive the rotation matrix for an angle \(\theta\)?
3. Why does rotating a vector by \(\frac{\pi}{2}\) result in a 90-degree counter-clockwise rotation?
4. How do you find the cosine and sine of special angles like \(\frac{\pi}{4}\) or \(\frac{\pi}{3}\)?
5. What are the properties of the rotation matrix?
6. Can you explain why the vector lengths remain the same after rotation?
7. How would you rotate a vector by a negative angle?
8. How does the rotation of vectors apply to real-world scenarios like computer graphics or physics?

**Tip:** Always verify the sign of sine and cosine functions based on the quadrant of the angle to ensure correct vector rotation.

Explore how to rotate a vector by different angles using matrix multiplication. Detailed step-by-step calculations for angles including π/4, π/2, 2π/3, 4π/3, 5π/3, and 7π/4. Match resulting vectors to correct angles. Suitable for advanced high school students or those studying vector mathematics.

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