Angle/Introduction/Section

Remark

For vectors $v$ and $w$, different from $0$, in a Euclidean vector space $V$, the inequality of Cauchy-Schwarz implies that

$-1\le \frac{⟨v,w⟩}{\Vert v\Vert \cdot \Vert w\Vert }\le 1$

holds. Using the trigonometric function cosine (as a bijective mapping $[0,\pi ]\to [-1,1]$) and its inverse function, the angle between the two vectors can be defined, by setting

$\mathrm{\angle }(v,w):=\arccos \frac{⟨v,w⟩}{\Vert v\Vert \cdot \Vert w\Vert }.$

The angle is a real number between $0$ and $\pi$. The equation above can be read as

$⟨v,w⟩=\Vert v\Vert \cdot \Vert w\Vert \cdot \cos (\mathrm{\angle }(v,w)).$

This provides the possibility to define the inner product in this way. However, then we have to find an independent definition for the angle. This approach might look a bit more intuitive but has many disadvantages, computationally and in terms of the proofs.

For an affine space $E$ over a Euclidean vector space $V$, and three given points $P,Q,R\in E$ (a triangle) with $Q,R\ne P$, the angle $\mathrm{\angle }(Q,P,R)$ of the triangle at $P$ is the angle $\mathrm{\angle }(\overrightarrow{PQ},\overrightarrow{PR})$.