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Scalar Triple Product

Let u and v b...

Question

Let $\to u$ and $\to v$ be the unit vectors such that $\to u \times \to v + \to u = \to w$ and $\to w \times \to u = \to v$ . Then the value of $[\to u \to v \to w]$ is equal to

A

- 1

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B

0

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C

1

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D

2

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Solution

The correct option is C $1$
Given: $\to u \times \to v + \to u = \to w \dots (i)$ and $\to w \times \to u = \to v \dots (i i)$
Multiplying equation $(i)$ by $\to u$
$(\to u \times \to v + \to u) \times \to u = \to w \times \to u$
$(\to u \times \to v \times \to u) + \to u \times \to u = \to v$ From $(i i)$
$(\to u \times \to v \times \to u) + 0 = \to v$
$(\to u . \to u) \to v - (\to u . \to v) \to u = \to v$
As $\to u$ is unit vector. So $\to u . \to u = 1$
$(1) \to v - (\to u . \to v) \to u = \to v$
$(\to u . \to v) \to u = 0$
Hence,
$(\to u . \to v) = 0 \dots (i i i)$
Now, $[\to u \to v \to w]$
$\Rightarrow \to u . (\to v \times \to w)$
$\Rightarrow \to u . (\to v \times (\to u \times \to v + \to u))$
$\Rightarrow \to u . (\to v \times (\to u \times \to v) + \to v \times \to u)$
$\Rightarrow \to u . ((\to v . \to v) \to u - (\to v . \to u) \to v + \to v \times \to u)$
$\Rightarrow \to u . ((1) \to u - 0 + \to v \times \to u)$
$\Rightarrow \to u . \to u + \to u . (\to v \times \to u)$
$\Rightarrow 1^{2} + [\to u \to v \to u]$
$\Rightarrow 1 + 0 = 1$

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Scalar Triple Product

Standard XII Mathematics

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