How do I simplify the following vectors:
a) a$\cdot$(a$\times$b),
b) (b-a)$\times$b and
c) (a-b)$\times$(b-a)
These are my efforts so far. I would appreciate if you could confirm if what I am doing is correct.
a) a$\cdot$(a$\times$b)= $0$
(Since there are two vectors, i.e. a which are equal in the scalar triple product, then its value is zero. Am I applying this property correctly?)
b) (b-a)$\times$b = (b$\times$b)-(a$\times$b)= $0$- a$\times$b= b$\times$a
Am I simplifying the above question correctly?
c) (a-b)$\times$(b-a)
(a-b)$\times$(b-a)
= (a$\times$b) - (b$\times$b) - [(a$\times$-a) - (-b$\times$-a)]
= (a$\times$b) - $0$ - (a$\times$-a) + (-b$\times$-a)
What do I do with all the negative signs?
Any help is welcome. Thanks
You can pull the minus sign in front. So you can simplify earlier: $$(\mathbf a-\mathbf b)\times(\mathbf b-\mathbf a)=-(\mathbf b-\mathbf a)\times(\mathbf b-\mathbf a)=0$$ The first two are correct