Question

If $3\overrightarrow{i}+2\overrightarrow{j}+8\overrightarrow{k}$ and $2\overrightarrow{i}+x\overrightarrow{j}+\overrightarrow{k}$ are at right angles, then $x=$

• A. 7
• B. -7
• C. 5
• D. -4

Answer 1

Answer: (B)

-7

The angle between $3\overrightarrow{i}+2\overrightarrow{j}+8\overrightarrow{k}$ and $2\overrightarrow{i}+x\overrightarrow{j}+\overrightarrow{k}$ is $\frac{\pi }{2}$
As $\cos \theta =\frac{|a_1{a}_2+b_1{b}_2+c_1{c}_2|}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}$
$\Rightarrow \cos \frac{\pi }{2}=\left|\frac{6+2x+8}{\sqrt{9+4+64}\sqrt{4+x^2+1}}\right|$

$\Rightarrow 2x=-14$

$\Rightarrow x=-7$