Why is the dot product the projection of one vector onto another? Could you please give me an intuitive explanation why the dot product is defined this way?
 A: It's not. The dot product is a mapping that takes two vectors and returns a scalar.
However, the dot product is connected to projections.
Take a vector $x\in V$ where $V$ is some vector space with an inner product over a field $\mathbb F$. Then that vector generates a linear subspace $V_x = \{\alpha x| \alpha\in\mathbb F\}$, and the projection onto this subspace (i.e., the mapping that projects each vector onto $V_x$) is defined as
$$P_x (y) = \langle x, y\rangle \frac{x}{\|x\|}.$$
A: Let's find the projection of a vector onto another analytically. We will define the projection as being the point on the supporting line of $\vec a$ which is the closest to the point $\vec b$.
In other words, we minimize the (squared) distance
$$d^2(\lambda\vec a,\vec b)=(\lambda a_x-b_x)^2+(\lambda a_y-b_y)^2+(\lambda a_z-b_z)^2.$$
This expression is a quadratic polynomial in $\lambda$ and we cancel the derivative with
$$\lambda(a_x^2+a_y^2+a_z^2)-(a_xb_x+a_yb_y+a_zc_z)=0.$$
This shows that for $\vec a$ being a unit vector,
$$\lambda=\vec a\cdot\vec b.$$
