Why is the dot product of two vectors a scalar value? I'm having some trouble seeing why dot products are said to give scalar values. As a far as I can see, it just gives another vector that is projected onto one of the 2 original vectors. How, then, is the result a scalar quantity. Can someone please explain this to me? Thank you. 

No, it doesn't give another vector. It gives the product of the length of one vector by the length of the projection of the other. This is a scalar.
You may have been misled by some figure.

The dot product is $|A|\,|B|\cos\theta$, not the vector $A'$.
