First slide

Scalar triple product of vectors

Question

$[(\vec{a} \times \vec{b}) \times (\vec{b} \times \vec{c}) (\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a}) (\vec{c} \times \vec{a}) \times (\vec{a} \times \vec{b})]$ is equal to (where $\vec{a}, \vec{b} and \vec{c}$ are non-zero non-coplanar vectors)

Moderate

A

$[\vec{a} \vec{b} \vec{c}]^{2}$
B

$[\vec{a} \vec{b} \vec{c}]^{3}$
C

$[\vec{a} \vec{b} \vec{c}]^{4}$
D

$[\vec{a} \vec{b} \vec{c}]$

Solution

$[(\vec{a} \times \vec{b}) \times (\vec{b} \times \vec{c}) (\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a}) (\vec{c} \times \vec{a}) \times (\vec{a} \times \vec{b})]$
$\begin{aligned} = [[\vec{a} \vec{b} \vec{c}] \vec{b} [\vec{a} \vec{b} \vec{c}] \vec{c} [\vec{a} \vec{b} \vec{c}] \cdot \vec{a}] \\ = {[\begin{array}{lll} a & b & c \end{array}]}^{3} [\vec{b} \vec{c} \vec{a}] = [\vec{a} \vec{b} \vec{c}]^{4} \end{aligned}$

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