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Vectors in Three Dimensions
A Euclidean vector is a geometric object that has magnitude (or length) and direction.
Learning Objectives

Define an Euclidean vector

Represent an Euclidean vector in the Cartesian coordinate system
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

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

tensor
a multidimensional array satisfying a certain mathematical transformation
Full Text
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
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 vectorlike 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
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Key Term Reference
 Cartesian
 Appears in this related concepts: Cylindrical and Spherical Coordinates, Double Integrals in Polar Coordinates, and Real Numbers, Functions, and Graphs
 Euclidean
 Appears in this related concepts: Partial Derivatives, Applications of Multiple Integrals, and Shape
 acceleration
 Appears in this related concepts: Motion with Constant Acceleration, Graphical Interpretation, and Applications and ProblemSolving
 algebraic
 Appears in this related concepts: Curve Sketching, The Dot Product, and Finding Limits Algebraically
 coordinate
 Appears in this related concepts: Parametric Equations, Calculus of VectorValued Functions, and Triple Integrals in Cylindrical Coordinates
 coordinate system
 Appears in this related concepts: Polar Coordinates, Conic Sections, and Surfaces in Space
 force
 Appears in this related concepts: Newton and His Laws, Work Done by a Variable Force, and Driven Oscillations and Resonance
 origin
 Appears in this related concepts: Adding and Subtracting Vectors Graphically, Overview of Muscle Functions, and ThreeDimensional Coordinate Systems
 vector
 Appears in this related concepts: VectorValued Functions, Plant Virus Life Cycles, and Superposition of Electric Potential
 velocity
 Appears in this related concepts: Rolling Without Slipping, Arc Length and Speed, and Tangent and Velocity Problems
Sources
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Cite This Source
Source: Boundless. “Vectors in Three Dimensions.” Boundless Calculus. Boundless, 02 Jul. 2014. Retrieved 14 Apr. 2015 from https://www.boundless.com/calculus/textbooks/boundlesscalculustextbook/advancedtopicsinsinglevariablecalculusandanintroductiontomultivariablecalculus5/vectorsandthegeometryofspace19/vectorsinthreedimensions1332849/