Question
If the straight lines having direction cosines given by al + bm + cn = 0 and fmn + gnI + hIm = 0 are perpendicular, then show that 
Solution
Given that a I + b m + c n = 0,
i.e.
...(1)
Also,
...(2)
Substituting value of n from (1) in (2), we get
or
On dividing both sides by m2, we have
If l1, m1, n1 and l2, m2 , n2 are the direction cosines of the two lines, then the roots of the equation (3) as
...(4)
Similarly, using (1) and (2) and by elimination of 1, we get
...(5)
Combining (4) and (5), we have
...(6)
We know that the lines with direction cosines l1, m1, n1 and l2, m2, n2 are perpendicular if
l1 l2 + m1m2 + n1n2 = 0 ....(7)
From (6) and (7), we get,
i.e.
...(1)Also,
...(2)Substituting value of n from (1) in (2), we get
or
On dividing both sides by m2, we have
If l1, m1, n1 and l2, m2 , n2 are the direction cosines of the two lines, then the roots of the equation (3) as
...(4)Similarly, using (1) and (2) and by elimination of 1, we get
...(5)Combining (4) and (5), we have
...(6)We know that the lines with direction cosines l1, m1, n1 and l2, m2, n2 are perpendicular if
l1 l2 + m1m2 + n1n2 = 0 ....(7)
From (6) and (7), we get,
