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Question: The vector <img src="https://cdn.pureessence.tech/canvas_119.png?top_left_x=300&top_left_y=1706&widt...

The vector , directed along the internal bisector of the angle between the vectors a=7i−4j−4k\mathbf { a } = 7 \mathbf { i } - 4 \mathbf { j } - 4 \mathbf { k } and b=−2i−j+2k\mathbf { b } = - 2 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } with , is

A

53(i−7j+2k)\frac { 5 } { 3 } ( \mathbf { i } - 7 \mathbf { j } + 2 \mathbf { k } )

B

53(5i+5j+2k)\frac { 5 } { 3 } ( 5 \mathbf { i } + 5 \mathbf { j } + 2 \mathbf { k } )

C

53(i+7j+2k)\frac { 5 } { 3 } ( \mathbf { i } + 7 \mathbf { j } + 2 \mathbf { k } )

D

53(−5i+5j+2k)\frac { 5 } { 3 } ( - 5 \mathbf { i } + 5 \mathbf { j } + 2 \mathbf { k } )

Answer

53(i−7j+2k)\frac { 5 } { 3 } ( \mathbf { i } - 7 \mathbf { j } + 2 \mathbf { k } )

Explanation

Solution

Let a\mathbf { a } = 7i−4j−4k7 \mathbf { i } - 4 \mathbf { j } - 4 \mathbf { k }and b=−2i−j+2k\mathbf { b } = - 2 \mathbf { i } - \mathbf { j } + 2 \mathbf { k }

Now required vector c=λ(a∣a∣+b∣b∣)\mathbf { c } = \lambda \left( \frac { \mathbf { a } } { | \mathbf { a } | } + \frac { \mathbf { b } } { | \mathbf { b } | } \right)

= λ(7i−4j−4k9+−2i−j+2k3)\lambda \left( \frac { 7 \mathbf { i } - 4 \mathbf { j } - 4 \mathbf { k } } { 9 } + \frac { - 2 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } } { 3 } \right) = λ9(i−7j+2k)\frac { \lambda } { 9 } ( \mathbf { i } - 7 \mathbf { j } + 2 \mathbf { k } )

̃ λ=±15\lambda = \pm 15 ̃ = ±53(i−7j+2k)\pm \frac { 5 } { 3 } ( \mathbf { i } - 7 \mathbf { j } + 2 \mathbf { k } )