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## Vector dot product

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### #1Alessandro  Members

Posted 13 January 2013 - 11:36 AM

I'm trying to use vector dot product to get the angle between two vectors, but I'm puzzled by the results.

Given two vectors, A and B:

A(0.0, 0.2, 0.0)
B(0.8, 0.0, 0.0)

The angle between those, calculated with the dot product:

properly returns 90°.

Now, let's have two vectors pointing in the same direction:

A(0.2, 0.2, 0.0)
B(0.8, 0.8, 0.0)

Would you guys explain me why in this case alpha is equal to 9.9°, while I'd expect to return 0°? Thanks

### #2Cornstalks  Members

Posted 13 January 2013 - 11:58 AM

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Your vectors aren't normalized. Normalize them first. That's required when using this equation.

Edit: (just to add an explanation for why the above matters)

To make it really clear, the dot product equation is:

A . B = ||A|| * ||B|| * cos(theta)

Where (A and B are your vectors, . is the dot product, and ||A|| means the magnitude of the vector A). In this form the equation, A and B are not required to be normalized (because the ||A|| * ||B|| part takes care of that... you'll see below).

Rearranging it, we see:

theta = acos((A . B) / (||A|| * ||B||))

Now, to normalize a vector A, you just do A / ||A||. I'll let An be the normlized version of A, where An = A / ||A||. Then you can plug it in as:

theta = acos(An . Bn)

Which is what you're trying to do, but your vectors aren't normalized, which is why you're getting problems.

So, to say it one more time:

theta = acos((A . B) / (||A|| * ||B||))   <-- General equation
theta = acos(An . Bn)   <-- Special case of the generalized equation (simplified because for normalized vectors An and Bn, ||An|| * ||Bn|| = 1)

Edited by Cornstalks, 13 January 2013 - 11:08 PM.

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