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Addition of Vectors

If a vector ...

Question

If a vector $\to r$ of magnitude $3 \sqrt{6}$ is directed along the bisector of the angle between the vectors $\to a = 7 ˆ i - 4 ˆ j - 4 ˆ k$ and $\to b = - 2 ˆ i - ˆ j + 2 ˆ k$ then $\to r =$

A

(ˆ i - 7 ˆ j + 2 ˆ k)

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B

(ˆ i + 7 ˆ j - 2 ˆ k)

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C

(- ˆ i + 7 ˆ j + 2 ˆ k)

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D

(ˆ i - 7 ˆ j - 2 ˆ k)

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Solution

The correct option is B $(ˆ i - 7 ˆ j + 2 ˆ k)$
Given

$\to a = 7^i - 4^j - 4^k$
$\to b = - 2^i -^j + 2^k$
angle bisector
$\to c = ∣ ∣ \to b ∣ ∣ \to a + | \to a | \to b$
$\to c = (\sqrt{(- 2)^{2} + (- 1)^{2} + 2^{2}}) (7^i - 4^j - 4^k) + (\sqrt{(7)^{2} + (- 4)^{2} + (- 4)^{2}}) (- 2^i -^j + 2^k)$
$\to c = 21^i - 12^j - 12^k - 18^i - 9^j + 18^k$
$\to c = 3^i - 21^j + 6^k$
unit vector of $\to c$
$^c = \frac{3^i - 2^j + 6^k}{9 \sqrt{6}}$
$\to r = m a g n i t u d e \times u n i t v e c t o r$
$\to r = 3 \sqrt{6} (\frac{3^i - 2^j + 6^k}{9 \sqrt{6}})$
$\to r =^i - 7^j + 2^k$

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directed along the bisector of the angle between

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