First slide

Scalar triple product of vectors

Question

$The volume of the tetrahedron formed by the coterminus edges \vec{a}, \vec{b}, \vec{c} is 3 .$ $Then the volume of the parallelepiped formed by the coterminus edges$ $\vec{a} + \vec{b}, \vec{b} + \vec{c}, \vec{c} + \vec{a} is$

Moderate

A

6
B

18
C

36
D

9

Solution

$\frac{1}{6} [\vec{a} \vec{b} \vec{c}] = 3 \Rightarrow [\vec{a} \vec{b} \vec{c}] = 18$
$Volume of the required parallelepiped$
$= [\vec{a} + \vec{b} \vec{b} + \vec{c} \vec{c} + \vec{a}] = |\begin{matrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{matrix}| [\vec{a} \vec{b} \vec{c}] = 2 [\vec{a} \vec{b} \vec{c}] = 36$

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