What does a dot product mean?

Question

I know the question seems to be a bit vague, but what do we actually do with the final result, eg I know the magnitude of 2 vectors and the angle between them, what is the result e get out of it? (I mean like even if one vector is 90deg to its initial position {cos(-x)}, I still get the same result, so how or where is this result used. Except Finding the norm of the vector. Also what does the inner product mean between 2 functions?

Answer 1

$\vec{a} \cdot \vec{b}$ Describes the component of $\vec{a}$ in the direction of $\vec{b}$ and vice versa. Thus qualitatively the dot product is a measure of how much two vectors lie in the same direction. As for inner product. You may be familiar that the dot product can be calculated using the formula $\vec{a}\cdot \vec{b} = \sum_{i=1}^n a_i b_i$ where $n$ is the number of components each vector has (usually 2 or 3). Inner product comes from imagining a function as a vector with infinite components. We go from a sum of descrete objects to a sum of continuous infinitly close objects ,thus we replace the $\sum$ with an $\int$. Thus the inner product of two functions $f(x)$ and $g(x)$ on $x \in [0,L]$ is $\int_0^L f(x)g(x)dx$.