# proof for $[\vec{a}\cdot (\vec{b} \times \vec{c})]\vec{a}=(\vec{a}\times\vec{b})\times(\vec{a}\times\vec{c})$

I encounter this triple product property in wikipedia But I can't find proof for $$[\vec{a}\cdot (\vec{b} \times \vec{c})]\vec{a}=(\vec{a}\times\vec{b})\times(\vec{a}\times\vec{c})$$
The RHS cross product produce vector while the LHS produce scalar.
So this got me stumble on working out this equation.
How do I get scalar equals to vector?
Does anyone know proof for this?

• I'm not sure what "The RHS cross product produce vector while the LHS produce dot product" means. Both sides of the equation are in fact vectors. – JavaMan Oct 30 '12 at 2:49
• Sorry,edited the text.Btw,I though dot product with vector always result in scalar. – kypronite Oct 30 '12 at 2:50
• The dot product of two vectors is a scalar. However you are taking the scalar $a \cdot (b \times c)$ and multiplying the vector $a$ by this scalar. – JavaMan Oct 30 '12 at 2:51
• sorry, you are correct.I overlooked that. – kypronite Oct 30 '12 at 2:53

$$(x \times y) \times z = (x \cdot z) y - (y \cdot z)x \\ x \cdot ( y \times z) = y \cdot (z \times x) = z \cdot (x \times y) \\ x \cdot (x \times y) = 0$$
We start with the righthand side. For convenience, denote $a \times c = v$. Then