The acute angle between two lines such that the direction cosines l,m,n of each of them sa…

Question

The acute angle between two lines such that the direction cosines $l,m,n$ of each of them satisfy the equations $l+m+n=0$ and $l^2+m^2-n^2=0$ is

Accepted Answer

From $l+m=-n$: $(l+m)^2=n^2$, and using $l^2+m^2=n^2$: $2lm=0\Rightarrow lm=0$, so each line has $l=0$ or $m=0$. Also $l^2+m^2-n^2=l^2+m^2+n^2-2n^2=1-2n^2=0\Rightarrow n=\pm\tfrac1{\sqrt2}$ for both lines.
$$\cos\theta=\frac12\implies\theta=60^{\circ}.$$

Solutions · 1