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What is the projection of a vector $u$ onto a nonzero vector $v$?
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The projection of a vector $u$ onto a nonzero vector $v$ is the vector denoted and defined by
$$\operatorname{proj}(u, v)=\frac{u \cdot v}{\|v\|^{2}} v=\frac{u \cdot v}{v \cdot v} v$$

This shows that this agrees with the usual notion of vector projection in physics.
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