Q:

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. 

A:

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 $A'$.

A:

$$(1,2)\cdot (3,4) = 1 (3) + 2(4) = 11$$
is a scalar.
I think you are confusing dot product with projection.
Suppose $u$ is a unit vector, we can project $v$ onto $u$ and its length would be $|u\cdot v|$ while the projection would be $(u\cdot v) u$.

