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

Represent an Euclidean vector in the Cartesian coordinate system

Define an Euclidean vector
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
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
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Key Term Reference
 Cartesian
 Appears in these related concepts: Cylindrical and Spherical Coordinates, Double Integrals in Polar Coordinates, and Real Numbers, Functions, and Graphs
 Euclidean
 Appears in these related concepts: Area of a Surface of Revolution, Partial Derivatives, and Shape
 acceleration
 Appears in these related concepts: Centripetial Acceleration, Position, Displacement, Velocity, and Acceleration as Vectors, and Graphical Interpretation
 algebraic
 Appears in these related concepts: Curve Sketching, Indeterminate Forms and L'Hôpital's Rule, and Finding Limits Algebraically
 coordinate
 Appears in these related concepts: Hyperbolic Functions, Physics and Engeineering: Center of Mass, and Parametric Equations
 coordinate system
 Appears in these related concepts: Polar Coordinates, Conic Sections, and Surfaces in Space
 force
 Appears in these related concepts: Newton and His Laws, Force, and Force of Muscle Contraction
 origin
 Appears in these related concepts: Adding and Subtracting Vectors Graphically, Overview of Different Muscle Functions, and ThreeDimensional Coordinate Systems
 vector
 Appears in these related concepts: Calculus of VectorValued Functions, Plant Virus Life Cycles, and Multiplying Vectors by a Scalar
 velocity
 Appears in these related concepts: Rolling Without Slipping, RootMeanSquare Speed, and Applications and ProblemSolving
Sources
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Cite This Source
Source: Boundless. “Vectors in Three Dimensions.” Boundless Calculus. Boundless, 21 Jul. 2015. Retrieved 30 Aug. 2015 from https://www.boundless.com/calculus/textbooks/boundlesscalculustextbook/advancedtopicsinsinglevariablecalculusandanintroductiontomultivariablecalculus5/vectorsandthegeometryofspace19/vectorsinthreedimensions1332849/