# A Nice Vector Cross Product Proof.

1. Feb 6, 2013

### Baumer8993

1. The problem statement, all variables and given/known data
If a, b, c, d are all vectors contained in the same plane, explain why
(a X b) X (c X d) = <0,0,0>

2. Relevant equations
The Cross Product!

3. The attempt at a solution

I know that since all of the vectors are in the same plane that means that a cross product between any of the vector will either be parallel, or anti-parallel. In this case the two cross products done create a vector with an angle of 180 degrees between them. So when they are crossed the result is 0. I just do not know how to "mathematically" prove this.

2. Feb 6, 2013

### Dick

I don't see anything wrong with the angle argument. But if you want something more algebraic, if all four vectors lie in the same plane then they are all linear combinations of two basis vectors for the plane, say u and v. Any ideas where to go from there?

3. Feb 6, 2013

### LCKurtz

Here's an alternate approach. Assuming you have the identity:$$\vec A \times (\vec C \times \vec D)=(\vec A \cdot \vec D)\vec C - (\vec A\cdot \vec C)\vec D$$Use that with $\vec A$ replaced by $\vec A \times \vec B$ and use what you know about the triple scalar (box) product of coplanar vectors.