Page 184 - Advanced engineering mathematics

164 CHAPTER 6 Vectors and Vector Spaces

These conclusions are proved by straightforward manipulations. Property (8) is proved by
3
a calculation identical to that done for vectors in R , thinking of F and G as vectors with n
components instead of three.
To verify property (9), use (8). First observe that the conclusion is just 0 ≤ 0 if either F
or G is the zero vector. Thus suppose both are nonzero. In property (8), choose α = G and
β =− F to obtain
2
0 ≤ αF + βG
2 2 2 2
= F G −2 F G (F · G)+ F G
= 2 F G ( F G −F · G).
Divide this inequality by 2 F G to obtain
F · G ≤ F G .

Now go back to conclusion (8) but this time set α = G and β = F to obtain, by a similar
computation,
2
0 ≤ αF + βG
= 2 F G ( F G +F · G).

Then
− F G ≤ F · G.
Put these two inequalities together to conclude that
− F G ≤ F · G ≤ F G ,

and this is equivalent to
|F · G|≤ F G .
There is no cross product for n-vectors if n > 3.
In view of the Cauchy-Schwarz inequality, we can define the cosine of the angle between
n-vectors F and G by

0 if F or G is the zero vector,
cos(θ) =
(F · G)/( F G ) if both vectors are nonzero.
This definition is motivated by the fact that this is the cosine of the angle between two vectors in
3
n
R . We use this definition to bring some geometric intuition to vectors in R , which we cannot
3
visualize if n > 3. For example, as in R , it is natural to define F and G to be orthogonal if their
dot product is zero (so the angle between the two vectors is π/2, or one or both vectors is the
zero vector).
If F and G are orthogonal, then F · G = 0. Upon setting α = β = 1 in (8) we obtain
2
2
2
F + G = F + G .
This is the n-dimensional version of the Pythagorean theorem.
n
Define standard unit vectors along the axes in R by
e 1 =< 1,0,0,··· ,0 >,
e 2 =< 0,1,0,··· ,0 >,··· ,
e n =< 0,0,··· ,0,1 >.
These vectors are orthonormal in the sense that each is a unit vector (length 1), and the vectors
are mutually orthogonal (each is orthogonal to all of the others).

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October 14, 2010 14:21 THM/NEIL Page-164 27410_06_ch06_p145-186