If the three coterminous edges of a parallele | vedclass

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(D) The volume $V$ of a parallelepiped with coterminous edges $\vec{u}, \vec{v}, \vec{w}$ is given by the scalar triple product $|[\vec{u}, \vec{v}, \

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(D) The volume $V$ of a parallelepiped with coterminous edges $\vec{u}, \vec{v}, \vec{w}$ is given by the scalar triple product $|[\vec{u}, \vec{v}, \vec{w}]| = |\vec{u} \cdot (\vec{v} 	imes \vec{w})|$.Here,the edges are $\vec{u} = (a - b)$,$\vec{v} = (b - c)$,and $\vec{w} = (c - a)$.Volume $V = |(a - b) \cdot ((b - c) 	imes (c - a))|$.Expanding the cross product: $(b - c) 	imes (c - a) = (b 	imes c) - (b 	imes a) - (c 	imes c) + (c 	imes a)$.Since $c 	imes c = 0$,we have $(b - c) 	imes (c - a) = (b 	imes c) - (b 	imes a) + (c 	imes a)$.Now,calculate the dot product with $(a - b)$:$V = (a - b) \cdot (b 	imes c - b 	imes a + c 	imes a)$$V = a \cdot (b 	imes c) - a \cdot (b 	imes a) + a \cdot (c 	imes a) - b \cdot (b 	imes c) + b \cdot (b 	imes a) - b \cdot (c 	imes a)$.Using the property of scalar triple products where any two vectors are identical,the result is $0$:$a \cdot (b 	imes c) = [a, b, c]$$a \cdot (b 	imes a) = 0$$a \cdot (c 	imes a) = 0$$b \cdot (b 	imes c) = 0$$b \cdot (b 	imes a) = 0$$b \cdot (c 	imes a) = [b, c, a] = [a, b, c]$.Substituting these: $V = [a, b, c] - 0 + 0 - 0 + 0 - [a, b, c] = 0$.

If the three coterminous edges of a parallelepiped are represented by the vectors $(a - b)$,$(b - c)$,and $(c - a)$,find its volume.

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