Dot Product of Two Vectors

If we have two vectors

and

that are in the same direction, then their dot product is simply the product of their magnitudes:

. To see this above, drag the head of

to make it parallel to

. If the two vectors are not in the same direction, then we can find the component of vector

that is parallel to vector

, which we can call

. and take the product of the magnitudes of

and

But how can we find

? If the angle between vectors

and

, then we can see that

, so

. Therefore, in general, we have that the dot product of

and

is:

where

is the angle between the two vectors

and

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