In this case, 0.6 squared equals 0.36;
0.8 squared equals 0.64. Both added
together with the zero equals 1. (If the
vector is already normalized, then the
value of |V| will be equal to one, and after
division the vector will remain as it was
before.)
How can vectorsbe representedin
various dimensional space?
Vectors can be found in two-, three-, or
multi-dimensional space. Two-dimen-
sional vectors are seen visually on a
graph as a line with an arrow connecting
two points. A two-dimensional vector is
defined by length and direction measured
by the angles that the arrow makes with
the xand ycoordinate system axes; a vec-
tor in such a coordinate system is written
as two components, (x, y).
Vectors in a three-dimensional space are represented with three numbers, one
along each coordinate axis. These are the coordinates of the arrow point, usually as
(x, y, z) if the arrow starts at the origin. A more complex vector is one with multiple
components, in which several different numbers in ordered n-tuples represent a vec-
tor. For example, (4, 1, 2, 0) is an ordered 4-tuple representing a vector in four
dimensions.
What is a polar coordinate system?
In effect, a polar coordinate system “wraps” a two-dimensional (Euclidean) coordinate
system onto the surface of a sphere (for more information about coordinate systems,
see “Geometry and Trigonometry”). A polar coordinate system examines a point in
space defined in terms of its position and distance on a sphere with a unit radius. The
center of the sphere is considered the origin; the first two coordinates are the longi-
tude and latitude on the sphere; the third coordinate defines the distance of the point
from the center of the sphere—the values of latitude, longitude, and height. In polar
coordinates—it’s easy to see on a globe of our own planet—latitude ranges from 90
to 90, longitude ranges from 180 to 180, and height ranges from zero to infinity.
Height can also be negative: The North Pole is at coordinates of (90, —, r) (the “—”
means there is no longitude), the South Pole is at coordinates of (90, —, r), and a
point on the equator is at coordinates (0, 0, r). 239