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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?

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    I'm not too sure what's being asked here; you appear to be asking multiple questions. Are you asking for the intuition behind a dot product? – 2017-02-13
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    Yes, that's what I wanted to ask. Also how would the dot product work with functions – 2017-02-14
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    I cannot provide insight for the dot products of functions, but I can give you insight for dot products of vectors. The dot product $ a * b $ is used to find the magnitude of $a$ in the direction of $b$. – 2017-02-14

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$\vec{a} \cdot \vec{b}$ 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 $\int_0^L f(x)g(x)dx$.

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    Could you please explain the inner product with an example with f(x) and g(x) as sin(x) and cos(x) respectively. (Not the mathematical calculations), but how would one define the magnitude of one with respect to the other. Thank you – 2017-02-14
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    Let's say we want to find the inner product of $sin(x)$ and $cos(x)$ on the interval $[0,2 \pi]$. As stated in in my answer $ = \int_0^{2 \pi}sin(x)cos(x)dx$ doing the integration you get that this is 0. Inner product of functions is usefull because it lets us think of functions as vectors and define ideas like orthogonality of functions (sin and cos are orthogonal on $[0,2\pi]$ since their inner product is 0). However, functions are not actually vectors and thus do not have properties like magnitude and direction. Is this example sufficient or are you still missing something? – 2017-02-14
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    Yes thank you, last question... Why do we need the dot product of two functions ( I think it is used in Fourier series) – 2017-02-14
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    We use dot product of functions to define function as "orthogonal" (perpendicular) to each other knowing that functions like sine and cosine are orthogonal allows you to cancel a lot of terms in Fourier series. In short it is a mathematical contrivance that is useful in picturing some problems. – 2017-02-14
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    One last thing, what I said before about functions not having a magnitude is not strictly true. The "magnitude" is called a norm and one possible norm (L-2 Norm) is the dot product of a function with itself. $|f(x)|_2 = $ . – 2017-02-14