First slide

Vector triple product

Question

If $(\vec{a} \times \vec{b}) \times (\vec{b} \times \vec{c}) = \vec{b}, where \vec{a}, \vec{b} and \vec{c}$ are non-zero vectors then

Moderate

A

$\vec{a}, \vec{b} and \vec{c}$ can be coplanar
B

$\vec{a}, \vec{b} and \vec{c}$ must be coplanar
C

$\vec{a}, \vec{b} and \vec{c}$ cannot be coplanar
D

none of these

Solution

$\begin{array}{l} (\vec{a} \times \vec{b}) \times (\vec{b} \times \vec{c}) = \vec{b} \\ or [\vec{a} \vec{b} \vec{c}] \vec{b} = \vec{b} \\ or [\vec{a} \vec{b} \vec{c}] = 1 \end{array}$
Therefore, $\vec{a}, \vec{b} and \vec{c}$ cannot be coplanar.

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