If vec{a}, vec{b}, vec{c} are three non-copla | vedclass

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(C) The volume of a parallelepiped formed by vectors $\vec{a}, \vec{b}, \vec{c}$ is given by the scalar triple product $|[\vec{a} \vec{b} \vec{c}]| =

Vedclass · Accepted Answer

$12$

Vedclass · Answer

(C) The volume of a parallelepiped formed by vectors $\vec{a}, \vec{b}, \vec{c}$ is given by the scalar triple product $|[\vec{a} \vec{b} \vec{c}]| = 4$. Thus,$[\vec{a} \vec{b} \vec{c}] = \pm 4$.We need to evaluate the expression:$E = (\vec{a} + \vec{b}) \cdot (\vec{b} 	imes \vec{c}) + (\vec{b} + \vec{c}) \cdot (\vec{c} 	imes \vec{a}) + (\vec{c} + \vec{a}) \cdot (\vec{a} 	imes \vec{b})$Using the property of scalar triple product $[\vec{x} \vec{y} \vec{z}] = \vec{x} \cdot (\vec{y} 	imes \vec{z})$,we can write:$E = [\vec{a} \vec{b} \vec{c}] + [\vec{b} \vec{b} \vec{c}] + [\vec{b} \vec{c} \vec{a}] + [\vec{c} \vec{c} \vec{a}] + [\vec{c} \vec{a} \vec{b}] + [\vec{a} \vec{a} \vec{b}]$Since the scalar triple product is zero if any two vectors are identical:$[\vec{b} \vec{b} \vec{c}] = 0, [\vec{c} \vec{c} \vec{a}] = 0, [\vec{a} \vec{a} \vec{b}] = 0$Also,by cyclic property $[\vec{a} \vec{b} \vec{c}] = [\vec{b} \vec{c} \vec{a}] = [\vec{c} \vec{a} \vec{b}] = V$Therefore,$E = V + 0 + V + 0 + V + 0 = 3V$Given $V = 4$,$E = 3 	imes 4 = 12$ (or $-12$ depending on orientation). Since $12$ is the provided option,the answer is $12$.

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