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Bisector of Angle between Two Vectors

Unit vector ...

Question

Unit vector $\to r$ which satisfies $\to r \times \to b = \to r \times \to c$ where $\to b = ˆ i + 2 ˆ j + ˆ k$ & $\to c = 3 ˆ i + 2 ˆ k$ , is

A

\pm (\frac{2 ˆ i - 2 ˆ j + ˆ k}{3})

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B

\pm (\frac{2 ˆ i + 2 ˆ j + ˆ k}{3})

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C

\pm (\frac{ˆ i + ˆ j + ˆ k}{\sqrt{3}})

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D

\pm ˆ i

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Solution

The correct option is B $\pm (\frac{2 ˆ i - 2 ˆ j + ˆ k}{3})$
Given $\to b = ˆ i + 2 ˆ j + ˆ k$ & $\to c = 3 ˆ i + 2 ˆ k$
$\to r \times \to b = \to r \times \to c$
$(\to r \times \to b) - (\to r \times \to c) = \to 0$
$\Rightarrow \to r \times (\to b - \to c) = \to 0$
$\Rightarrow \to r = λ (\to b - \to c) = λ (- 2 ˆ i + 2 ˆ j - ˆ k)$
$\Rightarrow ˆ r = \pm (\frac{2 ˆ i - 2 ˆ j + ˆ k}{3})$

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