Summary

Rest and motion only make sense relative to a chosen frame of reference; physics conventionally describes position with a right-handed Cartesian coordinate system. Physical quantities are either scalars (magnitude only) or vectors (magnitude and direction); vectors add via the triangle/parallelogram law, resolve into rectangular components, and combine either through the scalar (dot) product — giving a number, useful for angles and work — or the vector (cross) product — giving a new perpendicular vector, useful for areas, torque and angular quantities. The magnitude or norm of a vector $\vec A$ is $A=\sqrt{A_x^2+A_y^2+A_z^2}$; equal vectors have separately equal components; the position vector is $\vec r=x\hat i+y\hat j+z\hat k$.

Distance is the scalar path length travelled; displacement is the vector change in position. Average velocity is $\vec v_{avg}=\Delta\vec r/\Delta t$ and instantaneous velocity is $\vec v=d\vec r/dt$ — both vectors. Momentum is $\vec p=m\vec v$. Average acceleration is $\vec a_{avg}=\Delta\vec v/\Delta t$ and instantaneous acceleration is $\vec a=d\vec v/dt$. For constant acceleration, the four kinematic equations connect $u$, $v$, $a$, $s$, $t$, and specialise cleanly to free fall and vertical throw. …