Question

The value of $n$ so that vectors $2\hat{i}+3\hat{j}-2\hat{k},5\hat{i}+n\hat{j}+\hat{k}$ and $\hat{i}+2\hat{j}+3\hat{k}$ may be coplanar, will be:

• A. $66/8$
• B. $66/4$
• C. $66/2$
• D. $66$

Answer 1

The correct option is A $66/8$

$\overrightarrow{A}=2\hat{i}+3\hat{j}-2\hat{k}\overrightarrow{B}=5\hat{i}+n\hat{j}+\hat{k}\overrightarrow{C}=\hat{i}+2\hat{j}+3\hat{k}$

$\overrightarrow{A}$, $\overrightarrow{B}$ and $\overrightarrow{C}$ are coplanar if:

$\overrightarrow{B}\cdot (\overrightarrow{A}\times \overrightarrow{C})=0\Rightarrow (5\hat{i}+n\hat{j}+\hat{k})\cdot (4\hat{k}-6\hat{j}-3\hat{k}+9\hat{i}-2\hat{j}+4\hat{i})=0\Rightarrow (5\hat{i}+n\hat{j}+\hat{k})\cdot (13\hat{i}-8\hat{j}+\hat{k})=0\Rightarrow 65-8n+1=0\Rightarrow n=\frac{66}{8}$