What's the intuition for dot product as it is used in statistics? I know how to CALCULATE dot product, but still don't understand MEANING of it when it is being used in the context of statistics.
I mean, If I have to explain vector addition I would say that "Sum of two vectors give you the third vector that has the same effect as other two combined"
But I don't know how to explain dot product.
For example we have two vectors: v=(3,1) and w=(2,4). Dot product of these two is 32 + 14 = 10. What does this result (10) tell us?
 A: It is the sum of the covariance of the vectors, this is a measure of how strongly two vectors share their pattern of values. So in your final question the answer is that the two vectors share variance (covary) with a total magnitude of 10.
If the vectors are identical every value is squared, if they differ the result will be smaller than the square of the larger value. This means that the resulting value is useful for measuring how similar two vectors are.
If the two vectors are orthogonal they share no common variation and the sum of the products cancel out resulting in zero.
If the two vectors are unit vectors the maximum possible value is 1, the minimum is - 1,and it is the Pearson correlation coefficient for those two vectors.
If one vector is a unit vector then the result will be the magnitude of its contribution to the other vector. This is the basis of applying many modelling methods such as PCA and regression methods that use unit vectors as their core.
A: A probably more interesting way of formulating it is the following: 
$$
\langle v,w \rangle = \|v\|\|w\|\cos(\theta)
$$
where $\theta$ denotes the angle between the two vectors. (note that you have a certain way of understanding an angle in higher dimensions)
Rephrased, you have 
$$
\frac{\langle v,w \rangle}{\|v\|\|w\|} = \cos(\theta) \in [-1,1]. 
$$
In other words, the dot product gives you a measure of orthogonality (up to renormalization of the norms): the closer to $0$ the closer the two vectors are to being orthogonal. 
This is just one way to look at all this.
