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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?

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    sorry, you are correct.I overlooked that.2012-10-30

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All quantities below are vectors. I will use the following properties of cross-products and dot-products:

$ (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

\begin{align} (a \times b) \times (a \times c) = (a \times b) \times v &= (a \cdot v)b - (b \cdot v) a \\ &= (a \cdot(a \times c) )b - (b \cdot(a \times c)) a \\ &= 0 - (- a \cdot (b \times c))a \\ &= (a \cdot (b \times c) )a \end{align}

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    thanks for the workout.I had some geometry intuition after your initial commend but my vector algebra is sill elementary, difficulty piecing vector properties.2012-10-30