Mathematics

This is about decomposing a vector into its coordinate components using dot products. The key insight is that $\mathbf{i}$ and $\mathbf{j}$ form an orthonormal basis—each is a unit vector perpendicular to the other—so any 2D vector can be rebuilt by projecting it onto each axis and recombining. The dot product $\mathbf{u} \cdot \mathbf{i}$ extracts exactly the $x$-component of $\mathbf{u}$, and similarly for $\mathbf{j}$.

This is about decomposing a vector into its coordinate components using dot products. The key insight is that $\mathbf{i}$ and $\mathbf{j}$ form an orthonormal basis—each is a unit vector perpendicular to the other—so any 2D vector can be rebuilt by projecting it onto each axis and recombining. The dot product $\mathbf{u} \cdot \mathbf{i}$ extracts exactly the $x$-component of $\mathbf{u}$, and similarly for $\mathbf{j}$.