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

Practice representing 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
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
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
Vector in 3D Space
A vector in the 3D Cartesian space, showing the position of a point
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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: Partial Derivatives, Applications of Multiple Integrals, and Shape
 acceleration
 Appears in these related concepts: Centripetial Acceleration, Position, Displacement, Velocity, and Acceleration as Vectors, and Scientific Applications of Quadratic Functions
 algebraic
 Appears in these related concepts: Curve Sketching, The Dot Product, and Finding Limits Algebraically
 coordinate
 Appears in these related concepts: Parametric Equations, Calculus of VectorValued Functions, and Triple Integrals in Cylindrical Coordinates
 coordinate system
 Appears in these related concepts: Polar Coordinates, Conic Sections, and Surfaces in Space
 force
 Appears in these related concepts: Work Done by a Variable Force, Driven Oscillations and Resonance, and Force of Muscle Contraction
 origin
 Appears in these related concepts: Adding and Subtracting Vectors Graphically, Types of Muscle Tissue, and ThreeDimensional Coordinate Systems
 vector
 Appears in these related concepts: Introduction to Scalars and Vectors, Superposition of Electric Potential, and Introduction to Memory Storage
 velocity
 Appears in these related concepts: RootMeanSquare Speed, Tangent and Velocity Problems, and Force
Sources
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Cite This Source
Source: Boundless. “Vectors in Three Dimensions.” Boundless Calculus. Boundless, 20 May. 2016. Retrieved 24 May. 2016 from https://www.boundless.com/calculus/textbooks/boundlesscalculustextbook/advancedtopicsinsinglevariablecalculusandanintroductiontomultivariablecalculus5/vectorsandthegeometryofspace19/vectorsinthreedimensions1332849/