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Vector Triple Product

If x×y=a, y×z...

Question

If $\to x \times \to y = \to a, \to y \times \to z = \to b, \to x \cdot \to b = γ, \to x \cdot \to y = 1$ and $\to y \cdot \to z = 1$ . Vector $\to x$ is

A

\frac{1}{| \to a \times \to b |^{2}} [\to a \times (\to a \times \to b)]

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B

\frac{γ}{| \to a \times \to b |^{2}} [\to a \times \to b - \to a \times (\to a \times \to b)]

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C

\frac{γ}{| \to a \times \to b |^{2}} [\to a \times \to b + \to b \times (\to a \times \to b)]

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D

None of these

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Solution

The correct option is B $\frac{γ}{| \to a \times \to b |^{2}} [\to a \times \to b - \to a \times (\to a \times \to b)]$
$\to x \times \to y = \to a (i)$
$\to y \times \to z = \to b (i i)$
$\to x \cdot \to b = γ (i i i)$
$\to x \cdot \to y = 1 (i v)$
$\to y \cdot \to z = 1 (v)$

From $(i i)$ , $\to x \cdot (\to y \times \to z) = \to x \cdot \to b = γ$
$\Rightarrow [\to x \to y \to z] = γ$

From $(i)$ and $(i i)$ ,
$(\to x \times \to y) \times (\to y \times \to z) = \to a \times \to b$
$∴ [\to x \to y \to z] \to y - [\to y \to y \to z] \to x = \to a \times \to b$
or $\to y = \frac{\to a \times \to b}{γ} (v i)$

Also from $(i)$ , we get $(\to x \times \to y) \times \to y = \to a \times \to y$
$(\to x \cdot \to y) \to y - (\to y \cdot \to y) \to x = \to a \times \to y$

$\to x = \frac{1}{| \to y |^{2}} (\to y - \to a \times \to y)$

$= \frac{γ}{| \to a \times \to b |^{2}} ⎡ ⎢ ⎣ \frac{\to a \times \to b}{γ} - \frac{\to a \times (\to a \times \to b)}{γ} ⎤ ⎥ ⎦$

$= \frac{γ}{| \to a \times \to b |^{2}} [\to a \times \to b - \to a \times (\to a \times \to b)]$

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