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# done

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done

Mar 19, 2020
edited by rubikx2910  Apr 15, 2020

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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:

$$\begin{array}{|rcll|} \hline \mathbf{uv} &=& \|\mathbf{u}\|\|\mathbf{v}\|\cos{60^\circ} \\ \mathbf{uv} &=& 3*2*\dfrac12 \\ \mathbf{uv} &=& \mathbf{3} \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline \text{Let } \mathbf{u}&=&\dbinom{u_x}{u_y} \\ \text{Let }\mathbf{v}&=&\dbinom{v_x}{v_y} \\ \hline \mathbf{A} &=& \begin{pmatrix} u_x & u_y \\ v_x & v_y \end{pmatrix} \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline \mathbf{Au} &=& \begin{pmatrix} u_x & u_y \\ v_x & v_y \end{pmatrix}\dbinom{u_x}{u_y} \\\\ \mathbf{Au} &=& \dbinom{u_x^2+u_y^2}{u_xv_x+u_yv_y} \\\\ \mathbf{Au} &=& \dbinom{\|\mathbf{u}\|^2}{\mathbf{uv}} \\\\ \mathbf{Au} &=& \dbinom{3^2}{3} \\\\ \mathbf{Au} &=& \mathbf{\dbinom{9}{3}} \\ \hline \end{array} \begin{array}{|rcll|} \hline \mathbf{Av} &=& \begin{pmatrix} u_x & u_y \\ v_x & v_y \end{pmatrix}\dbinom{v_x}{v_y} \\\\ \mathbf{Av} &=& \dbinom{u_xv_x+u_yv_y}{v_x^2+v_y^2} \\\\ \mathbf{Av} &=& \dbinom{\mathbf{uv}}{\|\mathbf{v}\|^2} \\\\ \mathbf{Av} &=& \dbinom{3}{2^2} \\\\ \mathbf{Av} &=& \mathbf{\dbinom{3}{4}} \\ \hline \end{array}$$

Mar 19, 2020