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Vector dot product

Started by Alessandro Jan 13, 2013 at 5:36 PM 0 replies 1.6k views

Original Post

Jan 13, 2013 at 5:36 PM

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:

alpha= (acos(A*B)) * RAD_TO_DEG

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

Cornstalks

Jan 13, 2013 at 5:58 PM

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 A_n be the normlized version of A, where A_n = A / ||A||. Then you can plug it in as:

theta = acos(A_n . B_n)

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(A_n . B_n) <-- Special case of the generalized equation (simplified because for normalized vectors A_n and B_n, ||A_n|| * ||B_n|| = 1)

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