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# three dimensional vectors

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Let $$u$$ and $$v$$ be three-dimensional vectors such that the length of $$u$$ is $$2$$, the length of $$v$$ is $$4$$, and the angle between $$u$$ and $$v$$ when placed tail to tail is $$120^{\circ}$$.

Let $$\mathbf{A}$$ be a $$3\times 3$$ matrix such that

$$\mathbf{row}_1(\mathbf{A}) = \mathbf{u}, \mathbf{row}_2(\mathbf{A}) = \mathbf{v}, \mathbf{row}_3(\mathbf{A}) =3\mathbf{u} + 2\mathbf{v}.$$

Then what are $$\mathbf{Au}$$ and $$\mathbf{Av}$$ in that order? (Your answers should be numerical.)

Oct 24, 2021

$\mathbf{A} \mathbf{u} = \begin{pmatrix} 4 \\ -4 \\ 8 \end{pmatrix}$
$\mathbf{A} \mathbf{v} = \begin{pmatrix} -8 \\ 12 \\ -16 \end{pmatrix}$