Vectors¶
Recall (doc/postdoc). Not a first course.
Job: Euclidean structure on \(K^n\) (\(K=\mathbb{R},\mathbb{C}\)); geometric ops used in analysis and geometry.
Core¶
\[
v=(v_1,\ldots,v_n),\quad
\|v\|_2=\sqrt{\sum|v_i|^2},\quad
\langle u,v\rangle=\sum u_i\bar v_i\ \text{(sesquilinear convention varies)}
\]
| Op | Formula |
|---|---|
| Dot (\(\mathbb{R}\)) | \(u\cdot v=\|u\|\|v\|\cos\theta\) |
| Orthogonal | \(u\cdot v=0\) |
| Proj | \(\operatorname{proj}_v u=\dfrac{u\cdot v}{\|v\|^2}v\) |
| Cross (\(\mathbb{R}^3\)) | \(u\times v\); \(\|u\times v\|=\|u\|\|v\|\sin\theta\) |
| Line / plane | \(r_0+td\); \(n\cdot(r-r_0)=0\) |
Cauchy–Schwarz: \(|u\cdot v|\le\|u\|\|v\|\).
Used later for¶
- Vector spaces: abstract axiomatization
- Multivariable calculus: \(\nabla\), flux
- Functional analysis: Hilbert inner products
- Quantum / Fourier: \(L^2\) inner products
Check¶
Is the cross product associative on \(\mathbb{R}^3\)?
No — \((i\times i)\times j=0\) but \(i\times(i\times j)=i\times k=-j\) (up to standard basis). Lies in \(\mathfrak{so}(3)\) structure.