Question

Show that the three lines with direction cosines
$\frac{12}{13}$,$-\frac{3}{13}$,$-\frac{4}{13}$ ; $\frac{4}{13}$,$\frac{12}{13}$,$\frac{3}{13}$;$\frac{3}{13}$,$-\frac{4}{13}$,$\frac{12}{13}$ are mutually perpendicular.

Answer 1

Two lines with direction cosines l1, m1, n1 and l2, m2, n2 are perpendicular to each other, if l1l2+m1m2+n1n2=0

(i)For the lines with direction cosines, $\frac{12}{13}$, $-\frac{3}{13}$, $-\frac{4}{13}$ and $\frac{4}{13}$, $\frac{12}{13}$, $\frac{3}{13}$, we obtain
l1l2+m1m2+n1n2
=$\frac{12}{13}$×$\frac{4}{13}$+($-\frac{3}{13}$)×$\frac{12}{13}$+($-\frac{4}{13}$)×$\frac{3}{13}$
=$\frac{48}{169}$-$\frac{36}{169}$-$\frac{12}{169}$
=0

Thus, all the lines are mutually perpendicular.