Question

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

Answer 1

Answer: The above answer is correct.

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

Answer 2

Answers

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

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

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

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

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:

$A' = R\cdot A$ (1)

Where:

$A$ - Given vector matrix.

$R$ - Rotation matrix.

$A'$ - Resulting vector matrix.

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

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

Where $\theta$ is the Angle of Rotation, in radians.

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

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

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

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

$\theta = \frac{\pi}{4}$

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

$\theta = \frac{\pi}{2}$

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

$\theta = \frac{2\pi}{3}$

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

$\theta = \frac{4\pi}{3}$

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

$\theta = \frac{5\pi}{3}$

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

$\theta = \frac{7\pi}{4}$

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

Therefore, we have the following answers:

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

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

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

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

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