# Vectors in Three Dimensions

## A Euclidean vector is a geometric object that has magnitude (or length) and direction.

#### Key Points

• Vectors play an important role in physics.

• In the Cartesian coordinate system, a vector can be represented by identifying the coordinates of its initial and terminal point.

• The coordinate representation of vectors allows the algebraic features of vectors to be expressed in a convenient numerical fashion.

• Vectors can be added to other vectors according to vector algebra.

#### Terms

• a multidimensional array satisfying a certain mathematical transformation

• a quantity that transforms like a vector under a proper rotation but gains an additional change of sign under an improper rotation

#### Figures

1. ##### Vector in 3D Space

A vector in the 3D Cartesian space, showing the position of a point A represented by a black arrow. i,j,k are unit vectors in x,y,z axis, respectively

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 (Figure 1). For instance, in three dimensions, the points A = (1,0,0) and B = (0,1,0) in space determine the free vector $\overrightarrow{AB}$ 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{align} (1, 2, 3) + (−2, 0, 4) &= (1 − 2, 2 + 0, 3 + 4) \\ &= (−1, 2, 7) \end{align}.

#### Key Term Glossary

acceleration
the change of velocity with respect to time (can include deceleration or changing direction)
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algebraic
containing only numbers, letters, and arithmetic operators
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axis
a fixed, one-dimensional figure, such as a line or arc, with an origin and orientation and such that its points are in one-to-one correspondence with a set of numbers; an axis forms part of the basis of a space or is used to position and locate data in a graph (a coordinate axis)
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Cartesian
of or pertaining to co-ordinates based on mutually orthogonal axes
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coordinate
a number representing the position of a point along a line, arc, or similar one-dimensional figure
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coordinate system
a method of representing points in a space of given dimensions by coordinates from an origin
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Euclidean
adhering to the principles of traditional geometry, in which parallel lines are equidistant
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force
a physical quantity that denotes ability to push, pull, twist or accelerate a body which is measured in a unit dimensioned in mass × distance/time² (ML/T²): SI: newton (N); CGS: dyne (dyn)
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length
distance between the two ends of a line segment
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origin
the point at which the axes of a coordinate system intersect
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pseudovector
a quantity that transforms like a vector under a proper rotation but gains an additional change of sign under an improper rotation
vector
a directed quantity, one with both magnitude and direction; the signed difference between two points
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velocity
a vector quantity that denotes the rate of change of position with respect to time, or a speed with the directional component