Value of [a b c] from (a×(b×c)).(a×c)?

Question

If a and c are unit vectors at an angle $\pi/3$ with each other and (a×(b×c)).(a×c)=5, then what is the value of [a b c]? I just know the basic meaning of what are vector and scalar triple product. How do I do this question? I have to do it in about two minutes. Would someone please help?

Answer 1

We want to compute $a\cdot b\times c=-b\cdot a\times c$. We'll use $a\times (b\times c)=(a\cdot c)b-(a\cdot b)c$ so $$5=(a\times (b\times c))\cdot (a\times c)=(a\cdot c)(b\cdot a\times c).$$Hence $$[a b c]=\frac{-5}{a\cdot c}=\frac{-5}{\cos\frac{\pi}{3}}=-10.$$

Answer 2

Use $a\times(b\times c)=(a\cdot c)b-(a\cdot b) c$. Then $$(a\times(b\times c))\cdot(a\times c)=(a\cdot c)b\cdot(a\times c)-(a\cdot b)c\cdot(a\times c)=-(a\cdot b)[a,b,c]$$ etc.