Vectors in the Plane

A quantity with direction

A vector $\vec v=\langle a,b\rangle$ stores a magnitude and a direction in one object. Its length is $\lVert\vec v\rVert=\sqrt{a^2+b^2}$; its direction angle $\theta$ satisfies $\tan\theta=\frac{b}{a}$ (quadrant checked by the signs of $a,b$). Conversely, a vector of magnitude $r$ at angle $\theta$ has components $\langle r\cos\theta,\ r\sin\theta\rangle$.

Algebra: addition is componentwise, $\langle a,b\rangle+\langle c,d\rangle=\langle a+c,\ b+d\rangle$; a scalar stretches, $k\langle a,b\rangle=\langle ka,kb\rangle$, flipping direction when $k<0$. Geometrically, $\vec u+\vec v$ is the tip-to-tail diagonal — and $\lVert\vec u+\vec v\rVert$ is almost never $\lVert\vec u\rVert+\lVert\vec v\rVert$.

Dot product: $\vec u\cdot\vec v=u_1v_1+u_2v_2=\lVert\vec u\rVert\lVert\vec v\rVert\cos\theta$. It outputs a scalar and answers the angle question: positive means acute, zero means perpendicular, negative means obtuse.

Position vectors: a particle with position $\vec r(t)=\langle x(t),y(t)\rangle$ ties this topic back to parametrics — displacement from $t_1$ to $t_2$ is the single vector $\vec r(t_2)-\vec r(t_1)$.