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Inner product and projection


Here I’d like to highlight the relationship between a inner product of two vectors and projection of one vector on the other.

Let a and b be two vectors in \mathbb{R}^n. Then the projection of vector a on the vector b is \frac{bb^T}{b^Tb}a. The Euclidean length of the projection is \frac{b^Ta}{b^Tb}. This is well known. Based on this, we can get the following simple but useful fact.

  1. If \|a\|=1, then a^Tb or b^Ta is the projection length of b on a.
  2. If \|b\|=1, then a^Tb or b^Ta is the projection length of b on b.

This is quite straightforward, but it can often be encountered.


Edit: The above mentioned is orthogonal projection length. If you remember, this technique is used when derive the homography matrix. (N^Tx,\|N\|=1). In fact, a inner product of two vectors a,b is also a projection but not orthogonal projection (maybe Oblique projection).


\|a\|cos\theta and \|b\|cos\theta are orthogonal projections. But a^Tb=\|a\|\|b\|cos\theta is the oblique projection. One remarkable example is the Lie derivative.

\mathcal{L}_f h=\nabla h^T f

is the gradient of h along the vector f, which can be regarded as the projection of the gradient of h on the vector f.


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