+26367 Let \(\mathbf{u}\) and \(\mathbf{v}\) be vectors such that \(\|\mathbf{u}\| = 3\) and \(\|\mathbf{v}\| = 2\),
such that the angle between \(\mathbf{u}\) and \(\mathbf{v}\) when placed tail to tail is 60 degrees.
Let \(\mathbf{A}\) be a matrix such that \(\mathbf{row}_1(\mathbf{A}) = \mathbf{u},\ \mathbf{row}_2(\mathbf{A}) = \mathbf{v}\).
Then what are \(\mathbf{A} \mathbf{u},\ \mathbf{A} \mathbf{v}\)?
My attempt:
\(\begin{array}{|rcll|} \hline \mathbf{uv} &=& \|\mathbf{u}\|\|\mathbf{v}\|\cos{60^\circ} \\ \mathbf{uv} &=& 3*2*\dfrac12 \\ \mathbf{uv} &=& \mathbf{3} \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline \text{Let } \mathbf{u}&=&\dbinom{u_x}{u_y} \\ \text{Let }\mathbf{v}&=&\dbinom{v_x}{v_y} \\ \hline \mathbf{A} &=& \begin{pmatrix} u_x & u_y \\ v_x & v_y \end{pmatrix} \\ \hline \end{array} \)
\(\begin{array}{|rcll|} \hline \mathbf{Au} &=& \begin{pmatrix} u_x & u_y \\ v_x & v_y \end{pmatrix}\dbinom{u_x}{u_y} \\\\ \mathbf{Au} &=& \dbinom{u_x^2+u_y^2}{u_xv_x+u_yv_y} \\\\ \mathbf{Au} &=& \dbinom{\|\mathbf{u}\|^2}{\mathbf{uv}} \\\\ \mathbf{Au} &=& \dbinom{3^2}{3} \\\\ \mathbf{Au} &=& \mathbf{\dbinom{9}{3}} \\ \hline \end{array} \begin{array}{|rcll|} \hline \mathbf{Av} &=& \begin{pmatrix} u_x & u_y \\ v_x & v_y \end{pmatrix}\dbinom{v_x}{v_y} \\\\ \mathbf{Av} &=& \dbinom{u_xv_x+u_yv_y}{v_x^2+v_y^2} \\\\ \mathbf{Av} &=& \dbinom{\mathbf{uv}}{\|\mathbf{v}\|^2} \\\\ \mathbf{Av} &=& \dbinom{3}{2^2} \\\\ \mathbf{Av} &=& \mathbf{\dbinom{3}{4}} \\ \hline \end{array}\)
