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Definition of a Vector.

A vector is a mathematical object with magnitude and direction used to represent items such as the velocity, acceleration, or momentum of an object. A vector v can be represented by an n-tuple of real numbers:

v = [v_i] = (v₁, v₂,....., v_n)

considered to be elements (points) of Rⁿ, an n-dimensional real space.

If the n-tuples are complex numbers, then v is an element of Cⁿ, an n-dimensional complex space.

The 2-dimensional vector v = (v₁, v₂) in the diagram below has magnitude 12.81, the distance from the origin to (8, 10), and direction, the orientation of the arrow from the origin to (8, 10). Notice that the v₁ component is measured along the x-axis, while the v₂ component is measured along the y-axis.

Diagram a Vector v = [v₁, v₂]

v₁: 8; v₂: 10

The 3-dimensional vector a = (a₁, a₂, a₃) in the diagram below has magnitude 17.83, the distance from the origin to (13, 10, 7), and direction, the orientation of the line from the origin to (13, 10, 7). Notice that the a₁ component is measured along the x-axis, the a₂ component is measured along the y-axis, and the a₃ component is measured along the z-axis.

Конец формы

For example, the acceleration of an object at 17.83 meters per second² in the (13, 10, 7) direction can be represented by the line in the preceding diagram.