Hello, schteve!

Earboth showed you why the three cross products have the same *direction.*

Now we must prove they have the same *magnitude* (length).

The three vectors form a triangle. Code:

C
*
* *
|b| * * |a|
* *
* *
* *
A * * * * * * * B
|c|

Law of Sines: .

Multiply both sides by .[1]

Recall that the magnitude of a cross product is the area

. . of the parallelogram determined by the two vectors. Code:

* - - - - - - *
/:::::::::::::/
v /:::::::::::::/
/:θ:::::::::::/
* - - - - - - *
u

That is: .

Hence: .

According to [1], these two magnitudes are equal: .

In a similar fashion, we can prove that all three magnitudes are equal.