First slide

Vector triple product

Question

If $\vec{a} = 2 \hat{i} + \hat{j} + \hat{k}, \vec{b} = \hat{i} + 2 \hat{j} + 2 \hat{k}, \vec{c} = \hat{i} + \hat{j} + 2 \hat{k} and (1 + α) \hat{i} + β (1 + α) \hat{j} + γ (1 + α) (1 + β) \hat{k}$ $= \vec{a} \times (\vec{b} \times \vec{c})$ then $α, β and γ$ are

Moderate

A

$- 2, - 4, - \frac{2}{3}$
B

$2, - 4, \frac{2}{3}$
C

$- 2, 4, \frac{2}{3}$
D

$2, 4, - \frac{2}{3}$

Solution

$\begin{array}{l} \vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} \\ = 5 (\hat{i} + 2 \hat{j} + 2 \hat{k}) - 6 (\hat{i} + \hat{j} + 2 \hat{k}) \\ \Rightarrow (1 + α) \hat{i} + β (1 + α) \hat{j} + γ (1 + α) (1 + β) \\ \hat{k} = - \hat{i} + 4 \hat{j} - 2 \hat{k} \\ \Rightarrow 1 + α = - 1, β = - 4 and γ (- 1) (- 3) = - 2 \\ \Rightarrow γ = - \frac{2}{3} \end{array}$

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