Find the acute angle between the two straight lines whose direction cosines are given by `l+m+n=0` and `l^2+m^2-n^2=0`

Options

A. `(pi)/(3)`
B. `pi/4`
C. `pi/6`
D. `pi-2`

Answer

A

Explanation

We know that, angle between two line is `cos theta=(a_(1)a_(2)+b_(1)b_(2)+c_(1)c_(2))/(sqrt(a_(1)^(2)+b_(1)^(2)+c_(1)c^(2)sqrt(a_(2)^(2)+b_(2)^(2)+c_(2)^(2))))`
`therefore l+m+n=0 Rightarrow l=-(m+n)`
`Rightarrow (m+n)^(2)=l^(2)`
`Rightarrow m^(2)+n^(2)+2mn=m^(2)+n^(2)" "[therefore l^(2)=m^(2)+n^(2)]`
`Rightarrow 2mn=0`
when `m=0 Rightarrow l=-n`
Hence, (l,m,n) is (1,0,-1)
when, n=0, then l=-m
Hence, (l,m,n) is (1,0,-1) `therefore cos theta=(1+theta+0)/(sqrt2xxsqrt2)=1/2`
`Rightarrow theta=cos^(-1) ((1)/(2))=pi/3`