The volume of a parallelepiped whose cotermin | vedclass

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(D) The volume of a parallelepiped with coterminous edges $\vec{a}, \vec{b}, 	ext{ and } \vec{c}$ is given by the scalar triple product $|\vec{a} \cd

Vedclass · Accepted Answer

$4$

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(D) The volume of a parallelepiped with coterminous edges $\vec{a}, \vec{b}, 	ext{ and } \vec{c}$ is given by the scalar triple product $|\vec{a} \cdot (\vec{b} 	imes \vec{c})|$.This is equivalent to the absolute value of the determinant of the matrix formed by the components of the vectors:$V = |\det \begin{bmatrix} 2 & 1 & 1 \ 3 & -1 & 1 \ -1 & 1 & -1 \end{bmatrix}|$Expanding the determinant along the first row:$V = |2((-1)(-1) - (1)(1)) - 1((3)(-1) - (1)(-1)) + 1((3)(1) - (-1)(-1))|$$V = |2(1 - 1) - 1(-3 + 1) + 1(3 - 1)|$$V = |2(0) - 1(-2) + 1(2)|$$V = |0 + 2 + 2| = |4| = 4$Thus,the volume is $4$ cubic units.

The volume of a parallelepiped whose coterminous edges are represented by the vectors $\overrightarrow{OA} = (2, 1, 1)$,$\overrightarrow{OB} = (3, -1, 1)$,and $\overrightarrow{OC} = (-1, 1, -1)$ is . . . . . . cubic units.

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