The vector matrix [-3,-5] is rotated at different angles. Match the angles of rotation to the vector matrices they produce.

Answer: The above answer is correct.

Step-by-step explanation: I got this correct on Edmentum.

Answers

[tex]A' = \left[\begin{array}{cc}5.830\\-0.098\end{array}\right][/tex], [tex]\theta = \frac{2\pi}{3}[/tex]

[tex]A' = \left[\begin{array}{cc}-5.657\\-1.414\end{array}\right][/tex], [tex]\theta = \frac{7\pi}{4}[/tex]

[tex]A' = \left[\begin{array}{cc}-2.83\\5.098\end{array}\right][/tex], [tex]\theta = \frac{4\pi}{3}[/tex]

[tex]A' = \left[\begin{array}{cc}5\\-3\end{array}\right][/tex], [tex]\theta = \frac{\pi}{2}[/tex]

This exercise consist in finding the resulting Vector Matrix for each Angle of Rotation and a given Vector Matrix. The result is found by multiplying the given Vector Matrix for a Rotation Matrix, defined below as follows:

[tex]A' = R\cdot A[/tex] (1)

Where:

[tex]A[/tex] - Given vector matrix.

[tex]R[/tex] - Rotation matrix.

[tex]A'[/tex] - Resulting vector matrix.

For 2-dimension Vector Matrices, The Rotation Matrix is defined by the following entity:

[tex]R = \left[\begin{array}{cc}\cos \theta&-\sin \theta\\\sin \theta&\cos \theta\\\end{array}\right][/tex] (2)

Where [tex]\theta[/tex] is the Angle of Rotation, in radians.

Let be [tex]A = \left[\begin{array}{cc}x\\y\end{array}\right][/tex], the resulting vector matrix is found by (1) and (2):

[tex]A' = \left[\begin{array}{cc}\cos \theta&-\sin \theta\\\sin \theta&\cos \theta\end{array}\right] \cdot \left[\begin{array}{cc}x\\y\end{array}\right][/tex]

[tex]A' = \left[\begin{array}{cc}x\cdot \cos \theta - y\cdot \sin \theta\\x\cdot \sin \theta +y\cdot \cos \theta \end{array}\right][/tex]

If we know that [tex]A = \left[\begin{array}{cc}-3\\-5\end{array}\right][/tex], then the resulting vector matrix for each angle is, respectively:

[tex]\theta = \frac{\pi}{4}[/tex]

[tex]A' = \left[\begin{array}{cc}1.414\\-5.657\end{array}\right][/tex]

[tex]\theta = \frac{\pi}{2}[/tex]

[tex]A' = \left[\begin{array}{cc}5\\-3\end{array}\right][/tex]

[tex]\theta = \frac{2\pi}{3}[/tex]

[tex]A' = \left[\begin{array}{cc}5.830\\-0.098\end{array}\right][/tex]

[tex]\theta = \frac{4\pi}{3}[/tex]

[tex]A' = \left[\begin{array}{cc}-2.83\\5.098\end{array}\right][/tex]

[tex]\theta = \frac{5\pi}{3}[/tex]

[tex]A' = \left[\begin{array}{cc}-5.83\\0.098\end{array}\right][/tex]

[tex]\theta = \frac{7\pi}{4}[/tex]

[tex]A' = \left[\begin{array}{cc}-5.657\\-1.414\end{array}\right][/tex]

Therefore, we have the following answers:

[tex]A' = \left[\begin{array}{cc}5.830\\-0.098\end{array}\right][/tex], [tex]\theta = \frac{2\pi}{3}[/tex]

[tex]A' = \left[\begin{array}{cc}-5.657\\-1.414\end{array}\right][/tex], [tex]\theta = \frac{7\pi}{4}[/tex]

[tex]A' = \left[\begin{array}{cc}-2.83\\5.098\end{array}\right][/tex], [tex]\theta = \frac{4\pi}{3}[/tex]

[tex]A' = \left[\begin{array}{cc}5\\-3\end{array}\right][/tex], [tex]\theta = \frac{\pi}{2}[/tex]

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