Vectors in the Plane and in Space
A complete guide to understanding, computing with, and applying vectors in 2D and 3D.
1. Two Faces of a Vector
The Geometric Face
A vector is a quantity that has both a magnitude (length) and a direction. Geometrically, we picture it as a directed line segment — an arrow drawn from a starting point (the tail) to an ending point (the tip).
The Coordinate (Algebraic) Face
Algebraically, a vector is an ordered pair (in 2D) or ordered triple (in 3D) of real numbers:
In 2D: $\;\mathbf{v} = \langle a,\, b \rangle$ | In 3D: $\;\mathbf{v} = \langle a,\, b,\, c \rangle$
These numbers are called the components of the vector.
The Bridge: Position Vectors
The two views are connected by the position vector. If we place the tail of a vector at the origin $O$, its tip lands at a point whose coordinates are exactly the vector's components. So the point $P(a, b, c)$ corresponds to the position vector $\overrightarrow{OP} = \langle a, b, c \rangle$.
Both arrows have the same length and direction — they represent the same free vector $\langle 4, 3 \rangle$.