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)

$\left|\begin{array}{ccc}\lambda & \mu & 4\\ -2& 4& -2\\ 2& 3& 1\end{array}\right|=0$
$\lambda (10)=\mu (2)+4(-14)=0$
$10\lambda -2\mu =56$
$5\lambda -\mu =\mathrm{28.....}$(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)
$\Rightarrow \lambda +\mu =24$