Question

If the vectors $\overrightarrow{a}=2\hat{i}+\hat{j}+4\hat{k},\overrightarrow{b}=4\hat{i}-2\hat{j}+3\hat{k}$ and $\overrightarrow{c}=2\hat{i}-3\hat{j}-\lambda \hat{k}$ are coplanar, then the value of $\lambda$ is equal to

• A. 2
• B. 1
• C. 3
• D. $-1$

Answer 1

Answer: (B)

1

Answer 2

The correct option is(B): 1

$\because$ $\overrightarrow{a}=2\hat{i}+\hat{j}+4\hat{k},\overrightarrow{b}=4\hat{i}-2\hat{j}+3\hat{k}$ and $\overrightarrow{c}=2\hat{i}-3\hat{j}-\lambda \hat{k}$ are coplaner.
Hence, $\left| \begin{matrix} 2 & 1 & 4 \\\ 4 & -2 & 3 \\\ 2 & -3 & -\lambda \\\ \end{matrix} \right|=0$
$\Rightarrow$ $2(2\lambda +9)-1(-4\lambda -6)+(-12+4)=0$
$\Rightarrow$ $4\lambda +18+4\lambda +6-48+16=0$
$\Rightarrow$ $8\lambda -8=0$
$\Rightarrow$ $\lambda =1$