Question

If the angle between the lines whose direction cosines are $(-\frac{2}{\sqrt{21}},\frac{C}{\sqrt{21}},\frac{1}{\sqrt{21}})$ and $(\frac{3}{\sqrt{54}},\frac{3}{\sqrt{54}},-\frac{6}{\sqrt{54}})$ is $\frac{\pi }{2}$, then the value of $C$ is

• A. $6$
• B. $4$
• C. $-4$
• D. $2$

Answer 1

Answer: (B)

$4$

$(-\frac{2}{\sqrt{21}},\frac{C}{\sqrt{21}},\frac{1}{\sqrt{21}})$ and $(\frac{3}{\sqrt{54}},\frac{3}{\sqrt{54}},-\frac{6}{\sqrt{54}})$

must have product $=0$ as $cos{90}^o=0$

$cos\theta =x_1{x}_2+y_1{y}_2+z_1{z}_2=0$

$\frac{-6-6+3C}{\sqrt{2}1\times \sqrt{5}4}=0$

$\Rightarrow 3\times -2+3\times C+1\times -6=0\Rightarrow C=4$

Therefore Answer is $B$