Standard Basis Vectors of ℝ^q: Coordinates, Orthonormality and Unit Length

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $q\in\mathbb{N}$, let $[q]$ be as in Intervals of Natural Numbers §segment, and let points of $\mathbb{R}^{q}$ have components as in Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations §membership. For $k\in[q]$ let $e_{k}\in\mathbb{R}^{q}$ be the $k$-th standard basis vector, which exists and is unique by The Matrix Product, Matrix-Vector Product, Transpose, Trace, Identity Matrix and Standard Basis Vectors Exist and Are Unique §basis. For $x,y\in\mathbb{R}^{q}$ write $x\cdot y$ for the dot product and $\lVert x\rVert$ for the Euclidean norm. Let $k,l\in[q]$ and $w\in\mathbb{R}^{q}$.

¶ $e_{k}\cdot w=w_{k}$.

¶ $e_{k}\cdot e_{l}=1$ if $k=l$, and $e_{k}\cdot e_{l}=0$ if $k\neq l$.

¶ $\lVert e_{k}\rVert=1$.