DOT PRODUCT TEACH-O-MATIC

By Zach Barth

go to Cross Product Teach-O-Matic

Important stuff to take note of:

* Dot product is an area *

It can be calculated by either ||u|| * ||v|| * cos(theta) or ux*vx + uy*vy + uz*vz, which both work for random reasons. Using the ||u|| * ||v|| * cos(theta) method, we can conclude that the dot product ranges from 0 to the area of ||u|| * ||v|| depending on the value of the angle between the two vectors. In the simulation, dot product is represented by the red shaded area.

* How projections work *

projv u =  ( u·v ) v

        (||v||²)

Since u and v are both vectors, this means that (u·v) / (||v||^2) is a constant, meaning that (u·v) and (||v||^2) cancel out to leave a ratio which scales the vector v, creating a projected vector. In the simulation, the area of ||v||^2 is represented by the blue shaded area (some of which is covered by the dot product area, so note that ||v||^2 starts at orgin!). 

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