Question

A body is in equilibrium under the action of three force vectors A, B and C simultaneously.
Show that Ax B = BXC = CxĀ.​

Answer 1

Answer:

It's often convenient

to write the cross product as a determinant

i      j     k

Ax  Ay   Az

Bx  By    Bz

Taking just the i term

A X B = i [(Ay  Bz) - (Az By)]

B X C = i {(By Cz) - (Bz Cy)}

Now (A X B) - (B X C) = Bz (Ay + Cy) - By (Cz + Az)   (I)

For the body to be in equililbrium  (the vector components cancel)

Ax + Bx + Cx = Ay + By + Cy = Az + Bz + Cz = 0

Then  Ay + Cy = -By     and Cz + Az = -Bz

Substitute in (I)

- Bz By + Bz By = 0    since these are scalar quantities

The same is true for the j and k components so the differences of the vectors  A X B and B X C and C X A are zero and the vectors resultant cross products are equal