A Nice Vector Cross Product Proof.

[Baumer8993]

Homework Statement

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>

Homework Equations

The Cross Product!

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.

 

[Dick]

Homework Statement

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>

Homework Equations

The Cross Product!

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.

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?

 

[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.