First slide

Vector triple product

Question

if $\vec{a}, \vec{b} and \vec{c}$ are three unit vectors such that $\vec{a} \times (\vec{b} \times \vec{c}) = \frac{1}{2} \vec{b}$ ( $\vec{b} and \vec{c}$ being non-parallel)

Moderate

A

angle between $\vec{a} and \vec{b} is π / 3$
B

angle between $\vec{a} and \vec{c} is π / 3$
C

angle between $\vec{a} and \vec{b} is π / 2$
D

angle between $\vec{a} and \vec{c} is π / 2$

Solution

$\begin{array}{l} \vec{a} \times (\vec{b} \times \vec{c}) = \frac{1}{2} \vec{b} \\ or (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} = \frac{1}{2} \vec{b} \end{array}$
$\begin{array}{l} \Rightarrow \vec{a} \cdot \vec{c} = \frac{1}{2} and \vec{a} \cdot \vec{b} = 0 \\ \Rightarrow 1 \cdot 1 \cos α = \frac{1}{2} and \vec{a} ⊥ \vec{b} \\ \Rightarrow α = \frac{π}{3} and \vec{a} ⊥ \vec{b} \end{array}$

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