Question:
if l1,m1,n1 and l2,m2,n2 are the direction cosine of two mutually perpendicular lines show that the direction cosine of the line perpendicular to both of these are m1n2-m2n1, n1l2-n2l1,l1m2-l2m1
Answer:
Let u1 = (l1 , m1 , n1 ) is a unit vector along one line
and u2 = (l2 , m2 , n2 ) is a unit vector along the other line.
As u1 * u2 is a unit vector perpendicular to both u1 and u2
So, we calculate u1 * u2 as follows:
u1 * u2 = |i j k|
|l1 m1 n1 |
|l2 m2 n2 |
=> u1 * u2 = i(m1 n2 - m2 n1) - j(n2 l1 - n1 l2 ) + k(l1 m2 - l2 m1 )
=> u1 * u2 = i(m1 n2 - m2 n1) + j(n1 l2 - n2 l1 ) + k(l1 m2 - l2 m1 )
So, the required direction cosines are:
m1 n2 - m2 n1 , n1 l2 - n2 l1 , l1 m2 - l2 m1
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