Vectors & Coordinate Systems
Vectors, bases, dot/cross products — the algebraic backbone of physics.
A vector is a geometric object with magnitude and direction; algebraically it is a tuple of components in a chosen basis. In an orthonormal basis $\{\hat e_i\}$ of Euclidean space,
$$\mathbf v = \sum_i v_i \hat e_i, \qquad \mathbf v \cdot \mathbf w = \sum_i v_i w_i = |\mathbf v||\mathbf w|\cos\theta.$$The cross product in $\mathbb R^3$ produces a vector perpendicular to both inputs: $\mathbf v \times \mathbf w$ has magnitude $|\mathbf v||\mathbf w|\sin\theta$ and direction by the right-hand rule.
Under a rotation $R \in SO(3)$, components transform as $v'_i = R_{ij} v_j$. This transformation law is what distinguishes a true vector from an arbitrary triple of numbers — it's the entry point to tensor analysis.
Interactive: vector addition
Drag the sliders to change the two black/tan arrows. The red arrow is their sum.