Vector notation differences There is a admittedly naive question that is nagging me that I can't quite seem to nail down an answer for, probably because I don't know the right way to phrase the question. There are two notations of vector representation that I'm seeing in tutorials: $ a\vec{v} + b\vec{w} $ and $ a_x\hat{i} + a_y\hat{j} $ Are these representations able to express the same vector? Are they meant to be used interchangeably? I understand unit vectors and the notation fairly well but I'm not connecting the dots between these two representations.

They do not express the same vector per se (but they can of course by taking the unit vectors $\vec{v}=\vec{e}_1$ and $\vec{w}=\vec{e}_2$). $\vec{v},\vec{w}$ can express _any_ vector (let's work in the $2$-dimensional case), so $$a\vec{v} + b\vec{w}=a\begin{pmatrix} v_1 \\\ v_2 \end{pmatrix}+b\begin{pmatrix} w_1 \\\ w_2 \end{pmatrix}=\begin{pmatrix} av_1+bw_1 \\\ aw_2+bw_2 \end{pmatrix}$$ while $\hat{i}, \hat{j}$ express _unit_ vectors (because they represent standard basis vectors), so $$a_x\hat{i} + a_y\hat{j}=a_x\begin{pmatrix} 1 \\\ 0 \end{pmatrix}+a_y\begin{pmatrix} 0 \\\ 1 \end{pmatrix}=\begin{pmatrix} a_x \\\ a_y \end{pmatrix}$$