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.