Question

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

• A. 18
• B. 6
• C. 24
• D. 0

Answer 1

Answer: (C)

$ \begin{vmatrix} \lambda & \mu & 4 \\ -2 & 4 & -2 \\ 2 & 3 & 1 \end{vmatrix}=0$
$ \lambda(10)=\mu(2)+4(-14)=0 $
$ 10 \lambda-2 \mu=56$
$ 5 \lambda-\mu=28 .....$(1)
$ \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}=\sqrt{54}$
$\frac{-2 \lambda+4 \mu-8}{\sqrt{24}}=\sqrt{54} $
$ -2 \lambda+4 \mu-8=\sqrt{54 \times 24} .....$(2)
By solving equation (1) & (2)
$\Rightarrow \lambda+\mu=24$