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Boundless Calculus
Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus
Vectors and the Geometry of Space
Calculus Textbooks Boundless Calculus Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus Vectors and the Geometry of Space
Calculus Textbooks Boundless Calculus Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus
Calculus Textbooks Boundless Calculus
Calculus Textbooks
Calculus
Concept Version 9
Created by Boundless

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 AAA with a terminal point BBB, and denoted by AB⃗\vec{AB}​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)A=(1,0,0)A=(1,0,0) and B=(0,1,0)B=(0,1,0)B=(0,1,0) in space determine the free vector AB⃗\vec{AB}​AB​⃗​​ pointing from the point x=1x=1x=1 on the xxx-axis to the point y=1y=1y=1 on the yyy-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)O = (0,0,0)O=(0,0,0). Thus the bound vector represented by (1,0,0)(1,0,0)(1,0,0) is a vector of unit length pointing from the origin along the positive xxx-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)(1,2,3)(1,2,3) and $(−2,0,4)$ is the vector:

(1,2,3)+(2,0,4)=(12,2+0,3+4)=(1,2,7)\begin{aligned} (1, 2, 3) + (2, 0, 4) &= (1 2, 2 + 0, 3 + 4) \\ &= (1, 2, 7) \end{aligned}​(1,2,3)+(2,0,4)​​​​=(12,2+0,3+4)​=(1,2,7)​​.

Vector in 3D Space

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

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