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I can see why the dot product gives the angle between two vectors on $\mathbb {R}^{2}$, and that the angle between two vectors on $\mathbb {R}^{3}$ make sense, because you can take the plane defined by those two vectors, so it kind of falls back to $\mathbb {R}^{2}$, but what about $\mathbb {R}^{1}$ or $\mathbb {R}^{n}$ for $n \geq 4$? Is there such a thing as an angle in those other dimensions? I know there's such a thing as a distance, since by definition it's the length of a vector, but I can't grasp the concept of an angle then.

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    $\begingroup$ In a word, yes. $\endgroup$ – Lord Shark the Unknown Jul 29 '18 at 7:56
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    $\begingroup$ Maybe there are no angles in $n$ dimensions, but the dot product is still the product of the norms and the cosine of the angle ;-) $\endgroup$ – Yves Daoust Jul 29 '18 at 8:57
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In any dimension, any two vectors which are not collinear, span a plane. The angle between them is then exactly the same as the angle between two vectors in the plane.

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  • $\begingroup$ Is that plane always on the "second dimension", i.e., informally can we project those two vectors is 2D plane? $\endgroup$ – Luciano Santos Jul 29 '18 at 11:15
  • $\begingroup$ Yes, we can project these two vectors onto a two dimensional plane. $\endgroup$ – uniquesolution Jul 29 '18 at 11:55
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    $\begingroup$ @LucianoSantos: It's not just that you can project them onto a 2D plane. You can project anything onto anything you want. What is important here is that they lie in a two-dimensional plane. $\endgroup$ – tomasz Jul 29 '18 at 13:30
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The notion of an angle exists in a general inner product space for example (beyond $\mathbb {R}^n$ and the dot product). In the case of dot product on two or three tuples, the angle concept coincides with the 'geometric' concept that we are first familier with, from school for example.

The same is true about the notion of perpendicular or orthogonal. The orthogonal then is not necessarilly the same as perpendicular in the usual geometric sense.

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  • $\begingroup$ What does it measure, though? I mean, in a more abstract way. Some relationship between the vectors? $\endgroup$ – Luciano Santos Jul 29 '18 at 8:33
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    $\begingroup$ @Luciano Santos The concept of orthogonal is important in general. As in the basic case ('usual' perpendicular) orthogonal vectors are still useful when we try to mininise certain vector lengths. Again we can project orthogonaly to minimise the 'distance' bwn two vectors. Here I may be working with two ortogonal matrices, so I don't try to 'understand' the geometric 'meaning' but appreciate the algebraic result. $\endgroup$ – AnyAD Jul 29 '18 at 8:39

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