Question

The torque of a force →F about a point is defined as →Γ=→r×→F. Suppose →r, →F and →Γ are all nonzero. Is r×→Γ||→F always true? Is it ever true?

Answer 1

No its not true

Explanation:

it's never real.. The torque about a point of a force →F is never real.

That's because of the:→ r×→τ=→ r×→ r×→ Using a triple vector product, we get:→ r×→ r×→F=→ r.→F→ r-→ r.→ r→F∵→ r.→ r=r2=→ r.→F→ r-r2→FIf → r.→F=0; That's right.,  →r⊥→F, then:→ r×→Γ=-r2→F We know r2 will never be negative and → r×→Γ=-r2→.F It implies that both vectors can be antiparallel but not identical to each other.

Answer 2

Both vectors may be antiparallel to each other, but never parallel

Explanation:

→F about a point is defined as →Γ=→r×→F. Suppose →r, →F and →Γ are all nonzero. Is r×r→Γ||→F always true? Is it ever true?

r×r→Γ is not true. It could never be true because →r x r = →r x (→r x →F)

By using vector triple product, we get:

→r x (→r x →F) = (→r.F)→r - (→r.r)→F because →r.r = r^2.

= →r x →Γ= -r^2 →F

As r^2 is never negative and →r x →Γ = -r^2→F. So both vectors may be antiparallel to each other, but never parallel