# 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 and be two vectors in . Then the projection of vector on the vector is . The Euclidean length of the projection is . This is well known. Based on this, we can get the following simple but useful fact.

- If , then or is the projection length of on .
- If , then or is the projection length of on .

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. (). In fact, a inner product of two vectors is also a projection but not orthogonal projection (maybe Oblique projection).

and are orthogonal projections. But is the oblique projection. One remarkable example is the Lie derivative.

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

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