If the vectors a → = λ i ^ + μ j ^ + 4 k ^ , b → = - 2 i ^ + 4 j ^ - 2 k ^ and c…

If the vectors a→=λi^+μj^+4k^,b→=-2i^+4j^-2k^ and c→=2i^+3j^+k^ are coplanar and the projection of a→ on the vector b→ is 54 units, then the sum of all possible values of λ+μ is equal to
  1. 0
  2. 6
  3. 24
  4. 18

Solution

Given, $\vec{a}=\lambda\hat{i}+\mu\hat{j}+4\hat{k}$, $\vec{b}=-2\hat{i}+4\hat{j}-2\hat{k}$ and $\vec{c}=2\hat{i}+3\hat{j}+\hat{k}$ are coplanar. So, $\begin{vmatrix} \lambda & \mu & 4 \\ -2 & 4 & -2 \\ 2 & 3 & 1 \end{vmatrix}=0$ $\Rightarrow \lambda(10)-\mu(2)+4(-14)=0$ $\Rightarrow 10\lambda-2\mu=56$ $\Rightarrow 5\lambda-\mu=28$...(1) And given projection of $\vec{a}$ on $\vec{b}$ is $\sqrt{54}$ $\Rightarrow \frac{\vec{a}\cdot\vec{b}}{|\vec{b}|}=\sqrt{54}$ $\Rightarrow \frac{-2\lambda+4\mu-8}{\sqrt{24}}=\sqrt{54}$ $\Rightarrow -2\lambda+4\mu-8=\sqrt{54\times24}$...(2) By solving equation (1) and (2) we get, $\Rightarrow \lambda+\mu=24$

Asked in: JEE Main 2023 (29 Jan Shift 1)

Practice more Vectors questions on Aicharya