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Question

Volume of parallelepiped formed by vectors $$\vec{a} \times \vec{b}, \vec{b} \times \vec{c}$$ and $$\vec{c} \times \vec{a}$$ is 36 sq. units.


Solution

A: $$[\vec{a} \times \vec{b}  \vec{b} \times \vec{c}  \vec{c} \times \vec{a}] = 36$$
or $$[\vec{a}     \vec{b}    \vec{c}] = 6$$
B:
$$\Rightarrow$$ Volume of tetrahedron formed by vectors
$$\vec{a}, \vec{b}$$ and $$\vec{c}$$ is $$\displaystyle \dfrac{1}{6} [\vec{a} \vec{b} \vec{c}] = 1$$
C:
$$[\vec{a} + \vec{b}  \vec{b} + \vec{c}  \vec{c} + \vec{a} ] = 2 [\vec{a}\vec{b} \vec{c}] = 12$$
D:
$$\vec{a} - \vec{b}, \vec{b} - \vec{c} $$ and $$\vec{c} - \vec{a}$$ are coplanar
$$\Rightarrow [\vec{a} - \vec{b}    \vec{b} - \vec{c}    \vec{c} - \vec{a}] = 0$$

Maths

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