10.3#17

Good evening Dr. Taylor,

I am having trouble figuring out what the dot
product would be when looking at a graph; I realize that parallel and
orthogonal vectors would have a dot product of zero, but I can't tell when
two vectors would be positive or negative.

Thank you!
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ok, here's your hint.  the trick to this problem is exactly what we discussed in class on wednesday:

u.v = ||u|| ||v|| cos(θ)

Since u.v can be positive only if all of ||u|| ||v|| cos(θ) are positive, and can be negative only if both of  ||u||, ||v|| are positive (since they can't be negative) while cos(θ) is negative.  Looking at the vectors in the image, they are each clearly not the zero vector, hence in each case ||u||, ||v|| are positive. This means that the answer to all of these questions depends only on cos(θ).  So you can know everything you need to know by eye-balling the angles between each pair of vectors and knowing about the properties of cos(θ).  (NOTE: go study cos(θ))