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The dot product

1.

Another way to find the dot product is:

2.

Projections of a vector u onto a vector v are found in two ways I can see so far:

3.

Which is 100% equal to:

4.

WHERE is a unit vector pointing in the same direction as v.

SO we can also say:

5.

Where v has become a unit vector, totally.

Keep in mind the difference between equations 4 and 5 is in 4, v isn’t necessarily normalized when its used in the dot product operation, so you need to divided by |v| to “cancel its effect” on the final result. notice in all of these, we divide by v enough times so that the actual magnitude of v PLAYS NO ROLE in projvu, only the direction of v matters.

Finally you’ll see the following definition usually in physics books:

6.

Where t is the angle between vectors u and v.

This 6th formula follows if you study formula 3 and move things around a bit..

taking formula 3:

Notice that from formula 1, we have:

So, re-arranging formula 3 a bit now..

All I did there was add a |u| to the denominator, and separate the |v|2 into |v||v|

Ok?

Next, sub in

Then we get:

Now notice that , (v as a unit vector), so

where is a unit vector pointing in the direction of the vector v you are trying to project u onto.

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