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r
u ê ⋅ u
cos(γ ) = r m = m r (7.21)
m
u u
In 2-D or 3-D, these quantities have the geometrical interpretation of being
r
the cosine of the angles that the vector u makes with the x, y, and z axes.

7.3.3 Projections
r r
The projection of a vector u over a vector a is a vector whose magnitude is
r
r
the dot product of the vector u with the unit vector in the direction of , a
denoted by ê , and whose orientation is in the direction of ê :
a
a
r r r r r
⋅
⋅
r r ua a ua r
proj u () = ( u ê ê )⋅ = r r = a (7.22)
r
a a a r 2
a a a
r r
The component of u that is perpendicular to a is obtained by subtracting
r
r
r
from the projection vector of u u over . a
MATLAB Example
r r
Assume that we have the vector = ê + 5ê + 3ê + 7ê and the vector = 2êu 1 2 3 4 a 1
+ 3ê + ê + 4ê . We desire to obtain the components of each vector, the projec-
r
r
3
r
2
r
4
tion of u over , and the component of a u orthogonal to .a
Type, execute, and interpret at each step, each of the following commands
using the above definitions:
u=[1 5 3 7]
a=[2 3 1 4]
u(1)
a(2)
prjuovera=((u*a')/(norm(a)^2))*a
orthoutoa=u-prjuovera
prjuovera*orthoutoa'
The last command should give you an answer that is zero, up to machine
r
r
r
round-up errors because the projection of over and the component of u a u
r
orthogonal to a are perpendicular.
7.4 The Dirac Notation and Some General Theorems*

Thus far, we have established some key practical results in real finite dimen-
sional vector spaces; namely:

© 2001 by CRC Press LLC