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Applications of Dot Product

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Question

If $\to a$ and $\to b$ are non - zero vectors which are linearly dependent such that $\frac{| \to a + \to b |}{| \to a - \to b |} = 2, | \to b | > | \to a |$ then

A

\to b = 3 \to a

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B

\to b = - 3 \to a

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C

\to b = 2 \to a

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D

\to b = - 2 \to a

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Solution

The correct option is C $\to b = 3 \to a$
Given $\to a$ and $\to b$ are linearly dependent vectors.
So, let $\to b = λ \to a$
Consider, $\frac{| \to a + \to b |}{| \to a - \to b |} = 2$
$\Rightarrow \frac{| \to a + λ \to a |}{\to a - λ \to a} = 2$
$\Rightarrow \frac{1 + λ}{1 - λ} = \pm 2$
$\Rightarrow λ = 3, \frac{1}{3}$
$\Rightarrow \to b = 3 \to a (∵ | \to b | > | \to a |)$

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