Proving a basic vector equation

Question

I am trying to solve geometry problems and I have the following equation but I don't know how to prove it.

$$(a \times b)^2(a \times c)^2-((a \times b)(a \times c))^2 = a^2(abc)^2$$

Answer 1

By Pythagoras, the product of the squares of two vectors minus the square of their dot products, is the square of their cross product,

$$((a\times b)\times(a\times c))^2.$$

Then by the expulsion formula,

$$(((a\times b)\,c)\,a-((a\times b)\,a)\,c)^2=(a,b,c)^2a^2$$ where $(a,b,c)=(a\times b)\,c$ denotes the mixed product.