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  • If the vectors <img alt="\mathrm{\vec{a}=\lambda \hat{i}+\mu \hat{j}+4\hat{k},\vec{b}=2\hat{i}+4\hat{j}-2\hat{k}\: \: and\: \: \vec{c}=2\hat{i}+3\hat{j}+\hat{k}}" src="https://entrancecorner.oncode

If the vectors \mathrm{\vec{a}=\lambda \hat{i}+\mu \hat{j}+4\hat{k},\vec{b}=2\hat{i}+4\hat{j}-2\hat{k}\: \: and\: \: \vec{c}=2\hat{i}+3\hat{j}+\hat{k}} are coplanar and the projection of \mathrm{\vec{a}} on the vector \mathrm{\vec{b}} is \mathrm{\sqrt{54}} units, then the sum of all possible values of \mathrm{\lambda +\mu } is equal to

Option: 1

0


Option: 2

24


Option: 3

6


Option: 4

18


Answers (1)

best_answer

\mathrm{Vector\: \: \vec{a}=\lambda \hat{i}+\mu \hat{j}+4 \hat{k}, \vec{b}=-2 \hat{i}+4 \hat{j}-2 \hat{k} \: \: and\, \, \vec{c}=2 \hat{i}+3 \hat{j}+\hat{k}\: are\: \: coplanar\: \: then}

\begin{aligned} & {[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}]=0 \quad \Rightarrow\left|\begin{array}{ccc} \lambda & \mu & 4 \\ -2 & 4 & -2 \\ 2 & 3 & 1 \end{array}\right|=0} \\ & \end{aligned}

\begin{aligned} & \Rightarrow 10 \lambda-2 \mu-56=0 \\ & \end{aligned}

\begin{aligned} & \Rightarrow 5 \lambda-\mu=28 \\ & \end{aligned}  ....................(1)

also projection of \vec{a} on the \mathrm{\vec{b}\: is \: \sqrt{54}}  units. then

\begin{aligned} & \vec{a} \cdot \vec{b}=\sqrt{54} \\ & \Rightarrow \frac{-2 \lambda+4 \mu-8}{\sqrt{24}}=\sqrt{54} \\ & \Rightarrow-2 \lambda+4 \mu-8=36 \\ & \Rightarrow-2 \lambda+4 \mu=44 \end{aligned}..................(2)
from (1) and (2)

\begin{aligned} & \lambda=\frac{26}{3} \text { and } \mu=\frac{46}{3} \\ & \Rightarrow \lambda+\mu=\frac{26+46}{3}=\frac{72}{3}=24 \end{aligned}
 

Posted by

Divya Prakash Singh

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