The vector sum of a system of non-collinear forces acting on a rigid body is given to be non-zero. If the vector sum of all torques due to the system of forces about a certain point is found to be zero, does this mean that it is necessarily zero about any arbitrary point ?

Explanation

No, not necessarily true
Given `overset(N)underset(i) vecF_(i) ne0`
If `r_(i)` is the perpendicular distance from a certain point to the line of force then sum of torques about point O is
`vec tau overset(N) underset(i) sum =vect_(i)=overset(N) underset(i) sumvecr_(i)xx vecF_(i)=0`
The sum of torques about any other point is net perpendicular distance from point to the line of force
`vec tau= overset(N) underset(i)( vec r_(i)- veca)xx vecF_(i)`
or `vec tau= overset(N) underset(i) vecr_(i) xx vecF_(i) - vec axx underset(i) sum vecF_(i)`
So we find that, the second term on RHS may not vanish for any value of `veca`. Hence net torque about any arbitrary point O need not be necessarily zero.