Question

If l₁, m₁, n₁ and l₂, m₂, n₂ are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are m₁n₂ − m₂n₁, n₁l₂ − n₂l₁, l₁m₂ − l₂m₁.

Answer 1

It is given that l₁, m₁, n₁ and l₂, m₂, n₂ are the direction cosines of two mutually perpendicular lines. Therefore,

Let l, m, n be the direction cosines of the line which is perpendicular to the line with direction cosines l₁, m₁, n₁ and l₂, m₂, n₂,

Substituting the values from equations (5) and (6) in equation (4), we obtain