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.