Let a¯=αi^+2j^+βk^ and b¯=2i^−j^+5k^ be two vectors such that a¯×b¯=3i^−4j^−2k^ then the projection of b¯+3a¯ on b¯+a¯ is

|i^ j^ k^ α 2 β2 −1 5|=3i^−4j^−2k^ ⇒(10+β)i^+(2β−5α)j^+(−α−4)k^=3i^−4j^−2k^ ⇒β=−7; α=−2 b¯+3a¯=(2+3α)i^+(−1+6)j^+(5+3β)k^=−4i^+5j^−16k^ b¯+a¯=j^−2k^Projection of (b¯+3a¯) on (b¯+a¯)=5+325=375

Let a¯=αi^+2j^+βk^ and b¯=2i^−j^+5k^ be two vectors such that a¯×b¯=3i^−4j^−2k^ then the projection of b¯+3a¯ on b¯+a¯ is

|i^ j^ k^ α 2 β2 −1 5|=3i^−4j^−2k^ ⇒(10+β)i^+(2β−5α)j^+(−α−4)k^=3i^−4j^−2k^ ⇒β=−7; α=−2 b¯+3a¯=(2+3α)i^+(−1+6)j^+(5+3β)k^=−4i^+5j^−16k^ b¯+a¯=j^−2k^Projection of (b¯+3a¯) on (b¯+a¯)=5+325=375

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Let $\bar{a} = α \hat{i} + 2 \hat{j} + β \hat{k}$ and $\bar{b} = 2 \hat{i} - \hat{j} + 5 \hat{k}$ be two vectors such that $\bar{a} \times \bar{b} = 3 \hat{i} - 4 \hat{j} - 2 \hat{k}$ then the projection of $\bar{b} + 3 \bar{a}$ on $\bar{b} + \bar{a}$ is

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$\frac{14 \sqrt{5}}{3}$

$\frac{24 \sqrt{5}}{7}$

$\frac{37}{\sqrt{5}}$

$\frac{17}{\sqrt{5}}$

answer is D.

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Detailed Solution

$| \begin{array}{l} \hat{i} \hat{j} \hat{k} \\ α 2 β \\ 2 - 1 5 \end{array} | = 3 \hat{i} - 4 \hat{j} - 2 \hat{k} \Rightarrow (10 + β) \hat{i} + (2 β - 5 α) \hat{j} + (- α - 4) \hat{k} = 3 \hat{i} - 4 \hat{j} - 2 \hat{k} \Rightarrow β = - 7; α = - 2 \bar{b} + 3 \bar{a} = (2 + 3 α) \hat{i} + (- 1 + 6) \hat{j} + (5 + 3 β) \hat{k} = - 4 \hat{i} + 5 \hat{j} - 16 \hat{k} \bar{b} + \bar{a} = \hat{j} - 2 \hat{k}$
Projection of $(\bar{b} + 3 \bar{a})$ on $(\bar{b} + \bar{a}) = \frac{5 + 32}{\sqrt{5}} = \frac{37}{\sqrt{5}}$