length, norm concept
| Definition | The square root of the sum of a vector's squared coordinates, Pythagoras in d dimensions, which is also the square root of the vector's dot product with itself. Primer L, equations L.2 and L.3. |
|---|---|
| Example | (1, 2, 2) has length the square root of 1 + 4 + 4, which is 3. |
| Book | Data Mining as Observation, draft 0.2, commit f3914f0; entry id length-norm, kind concept. |
| Status | no ledger row names this entry. Corrections: none recorded. |
| Defining equation | Book equation L.2. |
| Assumptions and scope | none |
| Prior art | none recorded |
| Evidence | none |
| Reviewed | semantic review 2026-09-09; generated 2026-09-10 from records at the commits on the provenance page. |
Equation
Book equation L.2.
\[\|x\|=\sqrt{x_1^{2}+\cdots+x_d^{2}},\qquad u=\frac{x}{\|x\|},\qquad \|u\|=1.\]
Book equation L.3.
\[x\cdot y=x^{\top}y=\sum_{i=1}^{d}x_i y_i,\qquad x\cdot x=\|x\|^{2}.\]
Conditions
none
Ledger
none
First stated
Primer L section L.1 of Data Mining as Observation, added in draft 0.3 (2026-09-09) for the ECE 514 readers whose first courses are far behind. The idea is standard and TSK Appendix A covers it at length.
Measurements
none
Failures and corrections
none
Invariance envelope
none declared
Machine checked
none
Used in
Data Mining as Observation primers L and S, chapters 0, 1, 2, 3, 4, 6, 10, 11, 12, 13.
Related
unit vector; dot product; Euclidean distance; Frobenius norm.
See also
none
Status
Generated 2026-09-10 by encyclopedia/generate.py; book
at observation-data-mining f3914f0; the commit of every record is listed
in the encyclopedia’s provenance.