First slide

Scalar triple product of vectors

Question

Let $\vec{a}$ and $\vec{b}$ be unit vectors perpendicular to each other. Then $[\vec{a} + (\vec{a} \times \vec{b}) \vec{b} + (\vec{a} \times \vec{b}) \vec{a} \times \vec{b}]$ will always be equal to

Moderate

A

1
B

0
C

-1
D

none of these

Solution

$\begin{array}{l} [\vec{a} + (\vec{a} \times \vec{b}) \vec{b} + (\vec{a} \times \vec{b}) \vec{a} \times \vec{b}] \\ = (\vec{a} + (\vec{a} \times \vec{b})) \cdot ((\vec{b} + (\vec{a} \times \vec{b})) \times (\vec{a} \times \vec{b})) \\ = (\vec{a} + (\vec{a} \times \vec{b})) \cdot (\vec{b} \times (\vec{a} \times \vec{b})) \\ = (\vec{a} + (\vec{a} \times \vec{b})) \cdot (\vec{a} - (\vec{a} \cdot \vec{b}) \vec{b}) \\ = \vec{a} \cdot \vec{a} = 1 (as \vec{a} \cdot \vec{b} = 0, \vec{a} \cdot (\vec{a} \times \vec{b}) = 0) \end{array}$

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