Dot product

[ionic spire]

The dot product is supposed to represent vector similarity (the higher the dot product, the more similar the vectors). I don't understand how the dot product of (1 2 3) and (1 2 3) can be lower than the dot product (1 2 3) with (1 2 4) (or any 3d vector with larger positive numbers). Are they actually more similar than the vector (1 2 3) is to itself? What am I not understanding?

[distant mason]

1. Wait patiently for a helper to come along.
2. Once someone helps you, say thank you and close the thread with:

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3. Feel free to nominate the person for helper of the week in #helper-nominations

[barren tangle]

It isn't necessarily smaller.
The dot product is connected with the angle between vectors:
cos(θ) = (u·v)/(|u||v|)
Or, in terms of unit vectors:
e(u)·e(v) = cos(θ)
So, if you instead look at unit vectors, then the maximum value of the dot product will be when θ = 0, so when the vectors have the same direction.

[hexed lily]

They will represent vector "similarity" (for some notion of similarity) only if the vectors in question have the same length. Otherwise the dot product scales linearly with the length of the vector, so it won't be comparable in that case.

[ionic spire]

Ok, I understand. Thanks guys!

+close