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I am trying to prove that $\vec{A}=(\vec{A}\cdot \vec{n})\vec{n}+(\vec{n}\times\vec{A})\times\vec{n}$ where $\vec{n}$ is a unit vector and $\times$ indicates the cross product.

I am dealing with vectors in 3-dimensions in Klepner's book on mechanics, and so I assigned $\vec{A}$ in terms of $\hat{i}$, $\hat{j}$ and $\hat{k}$ and tried to do the same with the unit vector. That made my solution hideous.

I was wondering if someone could show me how to do the problem in $3$ dimensions elegantly.

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    There is no general cross product in $n$ dimensions.2012-12-05
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    There may still be a coordinate-free proof in 3 dimensions though, which I think is what the OP probably had in mind. But yes, it will be specific to dimension 3 so there actually is a cross product.2012-12-05
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    Yes, that is what I want.2012-12-05

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