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If $(3 \vec{a}-\vec{b}) \times(\vec{a}+3 \vec{b})=k \vec{a} \times \vec{b}$ then what is the value of $k$ ?
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The correct answer is:
10
$\quad \begin{aligned} (3 \overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}) \times(\overrightarrow{\mathbf{a}}+3 \overrightarrow{\mathbf{b}}) \\ &=(3 \overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}) \times \overrightarrow{\mathbf{a}}+(3 \overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}) \times 3 \overrightarrow{\mathbf{b}} \\ &=3 \overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{a}}+3 \overrightarrow{\mathbf{a}} \times 3 \overrightarrow{\mathbf{b}}-\overrightarrow{\mathbf{b}} \times 3 \overrightarrow{\mathbf{b}} \\ &=0-(-\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}})+9 \overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}-0 \\ &=10 \overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}} \\ & \therefore k=10 \end{aligned}$
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