If three vectors vec{a} = 12hat{i} + 4hat{j} | vedclass

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(D) The volume of a parallelepiped defined by vectors $\vec{a}, \vec{b},$ and $\vec{c}$ is given by the scalar triple product $|[\vec{a} \, \vec{b} \,

Vedclass · Accepted Answer

None of these

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(D) The volume of a parallelepiped defined by vectors $\vec{a}, \vec{b},$ and $\vec{c}$ is given by the scalar triple product $|[\vec{a} \, \vec{b} \, \vec{c}]|$.$V = |\vec{a} \cdot (\vec{b} 	imes \vec{c})| = \left| \det \begin{bmatrix} 12 & 4 & 3 \ 8 & -12 & -9 \ 33 & -4 & -24 \end{bmatrix} ight|$Expanding the determinant along the first row:$V = |12((-12)(-24) - (-9)(-4)) - 4((8)(-24) - (-9)(33)) + 3((8)(-4) - (-12)(33))|$$V = |12(288 - 36) - 4(-192 + 297) + 3(-32 + 396)|$$V = |12(252) - 4(105) + 3(364)|$$V = |3024 - 420 + 1092|$$V = |3696| = 3696 	ext{ cubic units.}$Since $3696$ is not among the options,the correct choice is $(d)$.

If three vectors $\vec{a} = 12\hat{i} + 4\hat{j} + 3\hat{k}$,$\vec{b} = 8\hat{i} - 12\hat{j} - 9\hat{k}$,and $\vec{c} = 33\hat{i} - 4\hat{j} - 24\hat{k}$ represent the coterminous edges of a parallelepiped,then its volume is:

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