Basis Vectors

$$\begin{gathered} B=\{{\vec{v}}_1,{\vec{v}}_2,{\vec{v}}_3\} \\ \text{ where }\Vert {\vec{v}}_i\Vert =1\text{ for }i=\mathrm{1...3} \\ \text{ and }{\vec{v}}_i\cdot {\vec{v}}_j=0\text{ for }i\ne j \end{gathered}$$

• B is an orthonormal set
  • orthogonal = dot product between all is 0 (all perpendicular to each other)
  • normal = norm is 1
• every vector can be uniquely expanded into 3 mutually perpendicular vectors that are parallel to the basis vectors

Standard Basis

• An orthonormal set of vectors where they all end up along the coordinate axises
• ${\hat{e}}_n$ has 0s in all spots except the $n^{th}$ spot where it has a 1 (this scales to infinite dimensions which is why we use this notation)

$${\hat{e}}_1=⟨1,0,0⟩,{\hat{e}}_2=⟨0,1,0⟩,{\hat{e}}_3=⟨0,0,1⟩$$ $$\vec{a}=a_1{\hat{e}}_1+a_2{\hat{e}}_2+a_2{\hat{e}}_2$$ $${\text{Proj}}_{{\hat{e}}_i}\vec{a}=a_i=({\vec{a}}_i\cdot {\hat{e}}_i){\hat{e}}_i=a_i{e}_i\text{ for }i=\mathrm{1...3}$$