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

Thank you for asking this question. Here is your answer:

Vectors along the given lines are  

(a↑) = ( l₁, m₁, n₁ ) and (b↑) = ( l₂, m₂, n₂ ).  

A line which is perpendicular to both (a↑) and (b↑)  

is along the cross product (a↑) x (b↑) =  

| i .. j .... k .|  

| l₁ ..m₁..n₁ | = ( m₁n₂ - m₂n₁ ) i - ( l₁n₂ - l₂n₁ ) j + ( l₁m₂ - l₂m₁ ) k ......... (1)  

| l₂ ..m₂..n₂ |  

Coefficients of i, j, k in (1) are the required d.C.s.