What purposes do the Dot and Cross products serve?
Do you have any clear examples of when you would use them?
Answer
When you deal with vectors, sometimes you say to yourself, “Darn I wish there was a function that…”

was zero when two vectors are perpendicular, letting me test perpendicularness.”
Dot Product

would let me find the angle between two vectors.”
Dot Product (actually gives the cosine of the angle between two normalized vectors)

would let me ‘project’ one vector onto another, or give the length of one vector in the direction of another.”
Dot Product

could tell me how much force is actually helping the object move, when pushing at an angle.”
Dot Product

could tell me how much a vector field is ‘spreading out’.”
Cross Product

could give me a vector that is perpendicular to two other vectors.”
Cross Product

could tell me how much torque a force was applying to a rotating system.”
Cross Product

could tell me how much this vector field is ‘curling’ up.”
Cross Product
There are actually a lot more uses, but the more I study vectors, the more and more I run into a situation where I need a function to do exactly something, and then realize that the cross/dot products already did exactly what I needed!
Attribution
Source : Link , Question Author : David McGraw , Answer Author : circular