Question

Find the direction cosines of the line which is perpendicular to the lines whose directionratios are 1, -1, 2 and 2, 1, -1

Answer 1

Let a,b and c be the dr's of the required line.
It is given that this line is perpendicular to the line whose dr's are 1,-1,2.
So, a(1)+b(-1)+c(2)=0
a-b+2c=0 ...(1)

It is given that this line is perpendicular to the line whose dr's are 2,1,-1.
So, a(2)+b(1)+c(-1)=0
2a+b-c=0 ...(2)

Solving (1) and (2),

a b c
1 -1 2
2 1 -1

$$\begin{gathered} \frac{a}{1-2}=\frac{b}{4+1}=\frac{c}{1+2}; \\ \frac{a}{-1}=\frac{b}{5}=\frac{c}{3}; \\ \sqrt{a^2+b^2+c^2}=\sqrt{(-1{)}^2+5^2+3^2}=\sqrt{1+25+9}=\sqrt{35} \\ Sodc\text{'}sare \\ \left(\frac{\pm a}{\sqrt{a^2+b^2+c^2}},\frac{\pm b}{\sqrt{a^2+b^2+c^2}},\frac{\pm c}{\sqrt{a^2+b^2+c^2}}\right)=\left(\frac{\pm 1}{\sqrt{35}},\frac{\pm 5}{\sqrt{35}},\frac{\pm 3}{\sqrt{35}}\right) \end{gathered}$$