Intuition behind the dot product

The dot product tells you what amount of one vector goes in the direction of another. This can also be interpreted as a similarity between the two vectors.

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The dot product of $\vec{a} = < a_1, a_2 >$ and $\vec{b} = < b_1, b_2 >$ can be represented as follows:

$ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \mathrm{cos}(\theta) $

$ \quad\quad = a_1 b_1 + a_2 b_2 $

$ \quad\quad = \vec{a} \vec{b}^T $