TITLE: Understanding Dot and Cross Product
QUESTION [53 upvotes]: What purposes do the Dot and Cross products serve? 
Do you have any clear examples of when you would use them?

REPLY [2 votes]: If you want me to name 2 concepts that are used in engineering calculations so frequently, they will be dot and cross products.There are several interpretations of the dot and cross product and can be applied in various scenarios: Angle between vectors, Projection of one vector in the direction of another as mentioned in the above posts.
Going a few steps above, interesting facts to remember are that the cross product of two vectors can be written as the product of a matrix(skew symmetric) and a vector.The norm of a cross product can be expressed as a determinant.
The triple product of three vectors namely a.(b$\times$c) represents the area of a parallelopiped and this triple product can also be written as a determinant.This useful fact can be used to prove collinearity of 3 points (of course after representing/considering them as vectors)