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The [dot] product may be understood geometrically as the projection of one vector onto another, multiplied by the length of the vector that it is projected onto. If one takes the dot product of two vectors $\vec{a}$ and $\vec{b}$, we can apply this procedure to find the correct formula for the dot product:

Let's call the angle between the vectors $\varphi$. Then, the projection of $\vec{a}$ onto $\vec{b}$ is $|\vec{a}|\cos \varphi$. Multiplying by the length of $\vec{b}$ gives us $$\vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\cos\varphi$$ This is the correct expression for the dot product. Of course, the dot product is symmetric so we might as well picture it as projecting $\vec{b}$ along $\vec{a}$. https://physics.stackexchange.com/a/111869/37286

• Vector Calculus: Understanding the Dot Product by Kalid Azad

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