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.