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What am I doing wrong here?

$(n^Tv)^2=(n^Tv)^T(n^Tv)=(v^Tn)(n^Tv)=v^T(nn^T)v=v^Tv$

$v$ is a 3d vector and $n$ is a 3d normal vector. From this calculation it looks like the projection length squared $(n^Tv)^2$ is independent of $n$, which is of course not true.

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$nn^T\ne1$, rather it is a $3\times 3$ matrix ($ne I_3$). – Tapu Jan 7 '12 at 11:25
up vote 2 down vote accepted

$nn^T\ne1$, rather it is a $3\times 3$ matrix ($\ne I_3$).

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In your equation $nn^T$ is an outer product and not an inner product. Therefore, you can't replace $v^T (nn^T) v$ with $v^Tv$.

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