Why is aX(bXc)=(a*c)b-(a*b)c reasonable? (vectors)

[Quisquis]

Homework Statement

Show that ax(bxc)=(a*c)b-(a*b)c. Why is it reasonable that ax(bxc) is some combination of b and c?

[The * indicates the dot product in this case because I can't find the appropriate latex symbol]

The Attempt at a Solution

I've shown the first part, but have no real idea how to describe the second. The best I've got so far:

It is reasonable because b and c are components of the final cross product with a.

That definitely seems lacking to me.

Any help on how to answer the 2nd part of the question would be much appreciated.

 

[lanedance]

why not just show it by expanding both sides?

for the resonable part, think of the direction of the vector (bxc) relative to b & c, then the direction of ax(bxc)

 

[Quisquis]

why not just show it by expanding both sides?

for the resonable part, think of the direction of the vector (bxc) relative to b & c, then the direction of ax(bxc)

I did expand both sides for the first part. I'm satisfied with that (it's a full page lol).

I know (bxc) is perpendicular to the plane defined by b and c.

Is the final resulting vector parallel to the plane defined by b and c (thinking about it, it seems like it would be)?