What is physical interpretation of dot product? Consider two vectors $V_1$ and $V_2$ in $\mathbb{R}^3$.
When we take their dot product we get a real number.
How is that number related to the vectors?
Is there any way we can visualize it? 

The dot product ${\bf a}\cdot{\bf b}$ measures the length of ${\bf a}$'s orthogonal projection onto $\bf b$ (the $1$-dimensional subspace it is a part of), scaled by the length of $\bf b$ itself. And conversely. The scaling is nice to have because it means the dot product is bilinear in its two arguments. Physically, it can measure how much of something in one direction moves in a different direction. For instance, suppose water is moving through a net (in the shape of a plane) put in the ocean - and, for simplicity, the water is moving monolithically: the same direction at the same speed at every point - but the direction of movement is not perfectly perpendicular to the net. The dot product of the net's unit normal vector with the water's velocity vector will tell us how much is moving through the net. When this is applied on an infinitessimal scale and then integrated, we can determine how much of something moving with a vector field is displaced through a surface; this is half of the divergence theorem.
