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Linear Dependence and Independence of Vectors

If b and c ar...

Question

If $\to b and \to c$ are two non-collinear unit vectors and $\to a$ is any vector, then $(\to a \cdot \to b) \to b + (\to a \cdot \to c) \to c + \frac{\to a \cdot (\to b \times \to c)}{| \to b \times \to c |^{2}} (\to b \times \to c) =$

A

\to a

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B

\to b

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C

\to c

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D

\to a + \to b + \to c

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Solution

The correct option is A $\to a$
Let $\to b =^i, \to c =^j$
$∴ | \to b \times \to c | = |^k | = 1$
again, let $a = a_{1}^i + a_{2}^j + a_{3}^k$
Now,
$\to a \cdot \to b = \to a \cdot^i = a_{1}, \to a \cdot \to c = \to a \cdot^j = a_{2}$
and $\to a \cdot \frac{\to b \times \to c}{| \to b \times \to c |} = \to a \cdot^k = a_{3}$
$∴ (\to a \cdot \to b) \to b + (\to a \cdot \to c) \to c + \to a \cdot \frac{\to b \times \to c}{| \to b \times \to c |} (\to b \times \to c)$
$= a_{1} \to b + a_{2} \to c + a_{3} (\to b \times \to c) = a_{1}^i + a_{2}^j + a_{3}^k = \to a$

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