91Ó°ÊÓ

Use a rotation matrix to rotate the vector \(\left[\begin{array}{l}-2 \\\ -3\end{array}\right]\) counterclockwise by the angle \(45^{\circ}\).

Find the parametric equation of the line in the \(x-y\) plane that goes through the indicated point in the direction of the indicated vector. $$(-1,-2),\left[\begin{array}{r}1 \\ -2\end{array}\right]$$

Use a rotation matrix to rotate the vector \(\left[\begin{array}{l}2 \\\ 1\end{array}\right]\) clockwise by the angle \(\pi / 3\).

Let $$\boldsymbol{A}=\left[\begin{array}{rr} -\mathbf{1} & \mathbf{1} \\ \mathbf{2} & \mathbf{3} \end{array}\right], \quad \boldsymbol{B}=\left[\begin{array}{ll} \mathbf{2} & \mathbf{0} \\ \mathbf{3} & \mathbf{2} \end{array}\right]$$ Find the inverse (if it exists) of \(A\).

Use a rotation matrix to rotate the vector \(\left[\begin{array}{l}1 \\\ 2\end{array}\right]\) counterclockwise by the angle \(\pi / 6\).

Let $$\boldsymbol{A}=\left[\begin{array}{rr} -\mathbf{1} & \mathbf{1} \\ \mathbf{2} & \mathbf{3} \end{array}\right], \quad \boldsymbol{B}=\left[\begin{array}{ll} \mathbf{2} & \mathbf{0} \\ \mathbf{3} & \mathbf{2} \end{array}\right]$$ Find the inverse (if it exists) of \(B\).

Find the parametric equation of the line in the \(x-y\) plane that goes through the indicated point in the direction of the indicated vector. $$(-1,4),\left[\begin{array}{l}2 \\ 3\end{array}\right]$$

Use a rotation matrix to rotate the vector \(\left[\begin{array}{r}5 \\\ -3\end{array}\right]\) clockwise by the angle \(\pi / 2\).

Find the parametric equation of the line in the \(x-y\) plane that goes through the given points. Then eliminate the parameter to find the equation of the line in standard form. \((-1,2)\) and \((3,0)\)

Let $$\boldsymbol{A}=\left[\begin{array}{rr} -\mathbf{1} & \mathbf{1} \\ \mathbf{2} & \mathbf{3} \end{array}\right], \quad \boldsymbol{B}=\left[\begin{array}{ll} \mathbf{2} & \mathbf{0} \\ \mathbf{3} & \mathbf{2} \end{array}\right]$$ Show that \(\left(A^{-1}\right)^{-1}=A\)