Question

The direction cosines to two lines at right angles are (1,2,3) and (-2,$\frac{1}{2}$,$\frac{1}{3}$), then the direction cosine perpendicular to both given lines are:

• A. $\sqrt{\frac{25}{2198}}$,$\sqrt{\frac{19}{2198}}$,$\sqrt{\frac{729}{2198}}$
• B. $\sqrt{\frac{24}{2198}}$,$\sqrt{\frac{38}{2198}}$,$\sqrt{\frac{730}{2198}}$
• C. $\frac{1}{3}$,-2,$\frac{-7}{2}$
• D. None of the above

Answer 1

The correct option is D None of the above

Direction cosines of $L_1=(1,2,3)$

Direction cosines of $L_2=(-2,\frac{1}{2},\frac{1}{3})$

Let direction ratios of required line be a,b,c then

$a+2b+3c=0\to 1$

and $-2a+\frac{b}{2}+\frac{c}{3}=0$

$\Rightarrow -12a+3b+2c=0\to 2$

Solving $1$ and $2$

$a+2b+3c=0$

$-12a+3b+2c=0$

$\Rightarrow \frac{a}{4-9}=\frac{b}{-36-2}=\frac{c}{3+24}$

$\Rightarrow \frac{a}{+5}=\frac{b}{+38}=\frac{c}{-27}$

$\Rightarrow a=+5,b=+38,c=-27$

$\therefore$Direction cosines are

$\frac{5}{\sqrt{5^2+{38}^2+{\left(27\right)}^2}},\frac{38}{\sqrt{5^2+{38}^2+{(-27)}^2}},\frac{-27}{\sqrt{5^2+{38}^2+{(-27)}^2}}$

$\frac{5}{\sqrt{2198}},\frac{38}{\sqrt{2198}},\frac{-27}{\sqrt{2198}}$