The volume of a parallelepiped whose cotermin | vedclass

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(C) The volume of a parallelepiped with coterminous edges $\vec{OA}, \vec{OB},$ and $\vec{OC}$ is given by the absolute value of the scalar triple pro

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$10$

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(C) The volume of a parallelepiped with coterminous edges $\vec{OA}, \vec{OB},$ and $\vec{OC}$ is given by the absolute value of the scalar triple product $[\vec{OA}, \vec{OB}, \vec{OC}]$.Given the coordinates $A(4, 3, 1), B(3, 1, 2),$ and $C(5, 2, 1)$,the vectors are $\vec{OA} = 4\hat{i} + 3\hat{j} + 1\hat{k}$,$\vec{OB} = 3\hat{i} + 1\hat{j} + 2\hat{k}$,and $\vec{OC} = 5\hat{i} + 2\hat{j} + 1\hat{k}$.The scalar triple product is calculated as the determinant:$[\vec{OA}, \vec{OB}, \vec{OC}] = \begin{vmatrix} 4 & 3 & 1 \ 3 & 1 & 2 \ 5 & 2 & 1 \end{vmatrix}$Expanding the determinant along the first row:$= 4(1(1) - 2(2)) - 3(3(1) - 5(2)) + 1(3(2) - 5(1))$$= 4(1 - 4) - 3(3 - 10) + 1(6 - 5)$$= 4(-3) - 3(-7) + 1(1)$$= -12 + 21 + 1 = 10$Since the volume must be positive,the volume is $|10| = 10$ cubic units.

The volume of a parallelepiped whose coterminous edges are represented by the vectors $\vec{OA}, \vec{OB},$ and $\vec{OC}$ with vertices $A(4, 3, 1), B(3, 1, 2),$ and $C(5, 2, 1)$ where $O$ is the origin,is ......... cubic units.

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