First slide

Scalar triple product of vectors

Question

If $\vec{a}, \vec{b} and \vec{c}$ be four non-zero vectors such that $[\vec{a} \vec{b} \vec{c}] = 1, \vec{c} = λ \vec{a} \times \vec{b}$ angle between $\vec{a} and \vec{b} is 2 π / 3, | \vec{a} | = \sqrt{2}$
$| \vec{b} | = \sqrt{3} and | \vec{c} | = \frac{1}{\sqrt{3}}$ the angle between $\vec{a} and \vec{b}$ is

Moderate

A

$\frac{π}{6}$
B

$\frac{π}{4}$
C

$\frac{π}{3}$
D

$\frac{π}{2}$

Solution

$\begin{array}{l} \vec{c} = λ (\vec{a} \times \vec{b}) \\ or \vec{c} \cdot \vec{c} = λ (\vec{a} \times \vec{b}) \cdot \vec{c} \\ \Rightarrow \frac{1}{3} = λ \\ Also | \vec{c} |^{2} = λ^{2} | \vec{a} \times \vec{b} |^{2} \\ \Rightarrow \frac{1}{3} = \frac{1}{9} (a^{2} b^{2} \sin^{2} θ) = \frac{1}{9} \times 2 \times 3 \sin^{2} θ \\ or \sin^{2} θ = \frac{1}{2} \\ or θ = \frac{π}{4} \end{array}$

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