Answer :
To find a line perpendicular to two given mutually perpendicular lines, a cross product can be used to obtain a resulting vector containing the direction cosines of the perpendicular line.
To show that the direction cosines of a line perpendicular to two given mutually perpendicular lines are l3, m3, and n3, we can use the vector cross product, as the resulting vector will be perpendicular to both original vectors.
If the first line has direction cosines l1, m1, and n1, and the second line has direction cosines l2, m2, and n2, then the cross product of their corresponding vectors A = l1i + m1j + n1k and B = l2i + m2j + n2k gives us a third vector C which is perpendicular to both A and B.
Thus, if we calculate the cross product A \\times B, we get C = (m1n2 - n1m2)i + (n1l2 - l1n2)j + (l1m2 - m1l2)k, the direction cosines of C, denoted by l3, m3, and n3 can be found by normalizing vector C.
