Question

I have a textbook that is doing something funny with cross products and I don't understand how/why they are allowed to do this, more info below momentarily.

Answer 1

Directly quoted from the text:

"Integration of the Equation of Motion.

You recall that the equation of motion for the two-body problem is
\[r''=-\frac{\mu}{r^3}r.\]Crossing this equation into h [a vector] leads toward a form which can be integrated:
\[r'' \times h =\frac{\mu}{r^3}(h \times r)"\]

Answer 2

Where mu is a constant, doesn't really matter, but the cross product isn't commutative, why the heck in that last equation is the order of the cross product flipped on the RHS???

Answer 3

If I crossed an entire equation, IDK, let's say \[ar = b\]with something else, c, where a, c, and c are vectors, I'd expect it to be\[(a \times c)r=(b \times c)\]

Answer 4

because, if we have two vectors, namely a, and b, due to the peoperty of cross product, we have:

Answer 5

\[a \times b=-b \times a\]
namely the cross product is anti-commutative

Answer 6

Oh, awesome, thank you. I totally missed that negative sign in the beginning and haven't had to apparently use that property in so long that I forgot...thank you!

Answer 7

thank you!