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7.3 Components, Direction Cosines, and Projections

7.3.1 Components
The components of a vector are the values of each element in the defining
r
n-tuplet representation. For example, consider the vector u = [1537]
in real 4-D. We say that its first, second, third, and fourth components are 1,
5, 3, and 7, respectively. (We are maintaining, in this section, the arrow nota-
tion for the vectors, irrespective of the dimension of the space.)
The simplest basis of a n-dimensional vector space is the collection of n unit
vectors, each having only one of their components that is non-zero and such
that the location of this non-zero element is different for each of these basis
vectors. This basis is not unique.
For example, in 4-D space, the canonical four-unit orthonormal basis vec-
tors are given, respectively, by:

ê = [1000] (7.15)
1
ê = [0100] (7.16)
2

ê = [0010] (7.17)
3
ê = [0001] (7.18)
4
r
and the vector can be written as a linear combination of the basis vectors:u

r
u = u ê + u ê + u ê + u ê (7.19)
1 1 22 33 4 4

The basis vectors are chosen to be orthonormal, which means that in addi-
tion to requiring each one of them to have unit length, they are also orthogonal
two by two to each other. These properties of the basis vectors leads us to the
following important result: the m component of a vector is obtained by tak-
th
ing the dot product of the vector with the corresponding unit vector, that is,

r
u = ê ⋅ u (7.20)
m m

7.3.2 Direction Cosines

The direction cosines are defined by:

© 2001 by CRC Press LLC