Vectors

Objective: Distinguish magnitude-only quantities from quantities with direction.

A scalar is described by a magnitude and unit, such as distance, speed, elapsed time, or mass. A vector includes magnitude and direction, such as displacement and velocity. In one dimension, a sign can encode direction relative to a chosen positive axis; in two dimensions, components or an angle are needed for full specification. Two equal-magnitude displacements in opposite directions add to zero, whereas the corresponding traveled distances add. Vector addition follows geometry and chosen coordinates, not simply addition of positive magnitudes. The units of displacement and distance are both length units despite their different meanings.

Objective: Apply vectors to calculations, evidence, and specified situations.

Represent a two-dimensional vector through perpendicular components. For angle θ from +x, x=r cosθ and y=r sinθ. Add corresponding signed components before finding resultant magnitude and direction; magnitudes alone do not generally add. For perpendicular components, magnitude is √(x²+y²). A diagram and stated axes prevent confusing an angle from horizontal with one from vertical. Opposite vectors negate both components.