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(B) The volume of a parallelepiped defined by three vectors $\vec{a}$,$\vec{b}$,and $\vec{c}$ is given by the scalar triple product: $V = |\vec{a} \cd

Vedclass · Accepted Answer

$12$

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(B) The volume of a parallelepiped defined by three vectors $\vec{a}$,$\vec{b}$,and $\vec{c}$ is given by the scalar triple product: $V = |\vec{a} \cdot (\vec{b} 	imes \vec{c})|$.This is equivalent to the absolute value of the determinant of the matrix formed by the components of the vectors:$V = \left| \det \begin{bmatrix} 1 & 2 & 0 \ 0 & 4 & 0 \ 0 & 1 & 3 \end{bmatrix} ight|$.Expanding the determinant along the first column:$V = |1(4 	imes 3 - 0 	imes 1) - 0 + 0| = |1(12) - 0 + 0| = 12$.Thus,the volume of the parallelepiped is $12$ cubic units.

The edges of a parallelepiped are represented by the vectors $\hat{i} + 2\hat{j}$,$4\hat{j}$,and $\hat{j} + 3\hat{k}$. Find its volume.

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