If and are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are

Solution:

Let’s consider be the direction cosines of the line perpendicular to each of the given lines.
Therefore,
And
On solving eq.(1) and eq.(2) by using cross – multiplication, we obtain

Therefore, the direction cosines of the given line are proportional to


Therefore, its direction cosines are

Where,

Where,

It is known to us that

Given that the given lines are perpendicular to each other.
Therefore,
Also, we have

And,
On substituting these values in eq. (3), we have

As a result, the direction cosines of the given line are