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Skill 5 of 14
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# Dot-product

Right now we have two versions of the work formula: $W=F_\parallel \Delta s$ and $W=F\,\Delta s_\parallel$, which include the $\parallel$-symbol, possibly some cosines when we have to calculate the components etc. All we wanted was to ‘multiply the parallel parts’. Let’s invent a simpler notation to mean just that:

$$W=\vec F\cdot \Delta \vec s$$

The well-known multiplication dot $\cdot$ is what we chose here to mean ‘multiply the parallel part of one vector onto the other vector’. Normal multiplication1 between numbers does not make sense between vectors,2 so we are simply reusing a symbol here that otherwise would have no meaning in relation to vectors. Why don’t we call this operation a dot-product.

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