Euclidean geometry on $\mathbb R^n$ uses $\langle x,y\rangle=x^Ty$. Complex coordinate geometry instead uses $\langle x,y\rangle=x^*y=\sum_j\overline{x_j}y_j$. In both cases $\|x\|=\sqrt{\sum_j|x_j|^2}$ and $d(x,y)=\|x-y\|$.
Orthogonality means inner product zero, and Cauchy–Schwarz gives $|\langle x,y\rangle|\le\|x\|\|y\|$. For nonzero real vectors, $\cos\theta=\langle x,y\rangle/(\|x\|\|y\|)$. Complex angle conventions must be specified separately.
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- Example of a sum of vectors interpreted geometrically in R^2In $\mathbb R^2$, take $u=(2,1)$ and $v=(-1,2)$. Their sum is $u+v=(1,3)$. Draw $v$ starting at the tip of $u$; the...
- Vector sum/addition interpreted geometrically in R^n (or C^n)Vector addition in $\mathbb R^n$ is componentwise and geometrically follows the head-to-tail or parallelogram rule....
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