Let a, b, c be three vectors that lie on the same line. Find \(\mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a}.\)

The vectors is $(1,0,0)$.

If two vectors are parallel, their cross product (vector product ) is zero.

So, all three products are zero in which case their sum will be zero.