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Dot product: Component vs. Geometric definition
The goal of this post is to answer a simple question: why are the following two definitions of the vector dot product in Euclidean space [1] equivalent for vectors \vec{a} and \vec{b}: Component definition: \vec{a}\cdot\vec{b}=\sum_{i=1}^{n}a_i b_i Geometric definition: \vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|cos(\theta), where |\vec{a}| is the magnitude of \vec{a} and is the angle between the vectors’ directions Here’s a graphical depiction of our vectors (focusing on for clarity, though this…
SRC Eli Bendersky