How can I prove $(a\times b)\cdot (c\times d) = (a\cdot c)(b\cdot d)-(a \cdot d)(b \cdot c)$?

Question

How can I prove $(a\times b)\cdot (c\times d) = (a\cdot c)(b\cdot d)-(a \cdot d)(b \cdot c)$? I already proved $a \times (b \times c) = (a \cdot c)b - (a \cdot b)c$, but is there anything that I can get from that proof? Do I have to laboriously try getting answer by putting $a = (a_{1}, a_{2}, a_{3})$, $b = (b_{1}, b_{2}, b_{3})$... and calculate that one by one?

Answer 1

Yes you can. Just additionally use the identity of the triple product: $$a\cdot(b \times c)= b \cdot(c \times a)$$ This leads to: $$(a\times b)\cdot (c\times d) = c \cdot (d \times (a\times b))$$ Now use the identity you already proved, and the desired result should follow.