Concepts Of Physics MCQ Edition [Volume 1]PhysicsPhysics and Mathematics
Given that the vectors A , B , and C are mutually perpendicular, evaluate C ( A B ) and determine whether the converse holds true.
Options
- AC ( A B ) = 0 , but the converse is not necessarily true.
- BC ( A B ) = 0 , and the converse is always true.
- CC ( A B ) 0 , and the converse is always true.
- DC ( A B ) 0 , but the converse is not necessarily true.
Correct answer
A. C ( A B ) = 0 , but the converse is not necessarily true.
Step-by-step solution
Using the vector triple product expansion: C ( A B ) = ( C B ) A - ( C A ) B Since A , B , and C are mutually perpendicular, the dot product of any two distinct vectors is zero: C B = 0 and C A = 0 Substituting these values into the expansion gives: C ( A B ) = 0 A - 0 B = 0 To check the converse, assume C ( A B ) = 0 . This implies: ( C B ) A - ( C A ) B = 0 If A and B are non-collinear, this equation requires C B = 0 and C A = 0 , which means C is perpendicular to both A and B . However, it places no restriction