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(A) Step $1$: Calculate the volume of the parallelepiped using the scalar triple product $\vec{a} \cdot (\vec{b} 	imes \vec{c})$.$\vec{b} 	imes \vec

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$2\sqrt{3}$

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(A) Step $1$: Calculate the volume of the parallelepiped using the scalar triple product $\vec{a} \cdot (\vec{b} 	imes \vec{c})$.$\vec{b} 	imes \vec{c} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 1 & 3 & -2 \ -3 & 1 & 4 \end{vmatrix} = \hat{i}(12 - (-2)) - \hat{j}(4 - 6) + \hat{k}(1 - (-9)) = 14\hat{i} + 2\hat{j} + 10\hat{k}$.Volume $V = |\vec{a} \cdot (\vec{b} 	imes \vec{c})| = |(-2)(14) + (5)(2) + (3)(10)| = |-28 + 10 + 30| = 12$.Step $2$: Calculate the area of the base formed by $\vec{b}$ and $\vec{c}$, which is $|\vec{b} 	imes \vec{c}|$.$|\vec{b} 	imes \vec{c}| = \sqrt{14^2 + 2^2 + 10^2} = \sqrt{196 + 4 + 100} = \sqrt{300} = 10\sqrt{3}$.Step $3$: The altitude $h$ is given by $V / 	ext{Area} = 12 / (10\sqrt{3}) = 6 / (5\sqrt{3}) = (6\sqrt{3}) / 15 = 2\sqrt{3} / 5$.

The parallelepiped is determined by vectors $\vec{a} = -2\hat{i} + 5\hat{j} + 3\hat{k}$, $\vec{b} = \hat{i} + 3\hat{j} - 2\hat{k}$, and $\vec{c} = -3\hat{i} + \hat{j} + 4\hat{k}$. The altitude of the parallelepiped on the parallelogram base determined by vectors $\vec{b}$ and $\vec{c}$ is

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