Please see the image attached.
Does that have anything to do with directions? The right-hand rule?
[itex]ca = ac[/itex], so yes.
Not in that figure, unless angle B is 90 degrees.
using the magnitude of the cross-product of two vectors to find the ar
If you mean
Area of triangle = (1/2)||A x C|| = (1/2)||C x A||
as in, using the magnitude of the cross-product of two vectors to find the area of the triangle between them, then yes.
I didn't see the second part of your question. The direction of ||A x C|| or ||C x A|| doesn't matter because they have the same magnitude. That is why (1/2)||A x C|| = (1/2)||C x A|| is true.
Thank you very much.
Separate names with a comma.