The direction cosines of two mutually perpendicular lines are given as and .
Since the lines are perpendicular, so,
Let the direction cosine of the line that is perpendicular to line with given direction cosines be . So,
The above equations can be written as,
The equations can be combined to a single equation as,
Since l, m and n are the direction cosines, so,
It is known that,
It is observed that,
On solving the above equation, the value comes out as,
Therefore, the direction cosines of the line are , and .