Question

if F is the force acting on a particle having position vector r and T the torque of the force about the origin then which condition is true?

Answer 1

T = r X F and 
r dot T 
= r dot (r X F) 
= (r X r) dot F 
[ In a Scalar Triple product of 3 vectors, dot and X are interchangeable 
i.e. A dot (B X C) = (A X B) dot C ] 
= 0 dot F = 0 
Similarly, 
F dot T 
= F dot (r X F) 
= r dot (F X F) 
[ In a Scalar triple product of 3 vectors, positions of the vectors can be interchanged cyclically 
i.e. A dot (B X C) = B dot (C X A) ] 
= r dot 0 = 0