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I've read that taking a dot product is just projecting one vector on the other, so a perpendicular vector will have no components in the other vectors direction. But shouldn't this leave the length unchanged so it has its original magnitude like multiplying it by 1?

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    $\begingroup$ "taking a dot product is just projecting one vector on the other" What? $\endgroup$ – Angina Seng Jan 4 at 20:50
  • $\begingroup$ When does the cosine rule reduce to the Pythagorean theorem? $\endgroup$ – Karl Jan 4 at 21:33
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$$u.v=|u||v|\cos \theta $$

If $ \theta =\pi/2$ we have $\cos \theta =0$

Thus perpendicular vectors have zero dot product.

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To clarify, the projection of $\vec u$ on $\vec v$ is the vector $$\left(\frac{\vec u\cdot \vec v}{\|\vec v\|^2}\right)\vec v = \left(\frac{\vec u\cdot\vec v}{\|\vec v\|}\right)\frac{\vec v}{\|\vec v\|}.$$ The dot product is a scalar quantity. But the length of the projection is always strictly less than the original length unless $\vec u$ is a scalar multiple of $\vec v$.

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The dot product is that way by definition, this particular definition gives the expected Euclidean Norm. A consistent dot product can be and is defined differently, for example in physics & differential geometry the metric tensor is solved for and ascribes a different inner product at every space-time coordinate, which is the means for modeling curved spaces.

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