Vectors in Three Dimensions

A Euclidean vector (sometimes called a geometric or spatial vector, or—as here—simply a vector) is a geometric object that has magnitude (or length) and direction and can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point $A$ with a terminal point $B$, and denoted by $\vec{AB}$.

Vectors play an important role in physics: velocity and acceleration of a moving object and forces acting on it are all described by vectors. Many other physical quantities can be usefully thought of as vectors. The mathematical representation of a physical vector depends on the coordinate system used to describe it. Other vector-like objects that describe physical quantities and transform in a similar way under changes of the coordinate system include pseudovectors and tensors.

In the Cartesian coordinate system, a vector can be represented by identifying the coordinates of its initial and terminal point. For instance, in three dimensions, the points $A=(1,0,0)$ and $B=(0,1,0)$ in space determine the free vector $\vec{AB}$ pointing from the point $x=1$ on the $x$-axis to the point $y=1$ on the $y$-axis. 

Typically in Cartesian coordinates, one considers primarily bound vectors. A bound vector is determined by the coordinates of the terminal point, its initial point always having the coordinates of the origin $O = (0,0,0)$. Thus the bound vector represented by $(1,0,0)$ is a vector of unit length pointing from the origin along the positive $x$-axis. The coordinate representation of vectors allows the algebraic features of vectors to be expressed in a convenient numerical fashion. For example, the sum of the vectors $(1,2,3)$ and $(−2,0,4)$ is the vector:

$\begin{aligned} (1, 2, 3) + (2, 0, 4) &= (1 2, 2 + 0, 3 + 4) \\ &= (1, 2, 7) \end{aligned}$.

Vector in 3D Space

A vector in the 3D Cartesian space, showing the position of a point $A$ represented by a black arrow. $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ are unit vectors along the $x$-, $y$-, and $z$-axes, respectively.