Page 175 - Advanced engineering mathematics

6.2 The Dot Product 155

Dot products of vectors can be computed using MAPLE and the DotProduct command,
which is in the VectorCalculus package of subroutines. This command also applies to n-
dimensional vectors, which are introduced in Section 6.4.
Conclusions (1), (2), and (3) are routine computations. Conclusion (4) is often used in
computations. To verify conclusion (4), suppose
F = ai + bj + ck.

Then
2
2
2
2
F · F = a + b + c = F .
Conclusion (5) follows easily from (4), since O is the only vector having length 0. For conclusion
(6), use conclusions (1) through (4) to write
2
αF + βG = (αF + βG) · (αF + βG)
2
2
= α F · F + αβF · G + αβG · F + β G · G
2
2
2
2
= α F +2αβF · G + β G .
The dot product can be used to find an angle between two vectors. Recall the law of cosines:
For the upper triangle of Figure 6.11 with θ being the angle opposite the side of length c,thelaw
of cosines states that
2
2
2
a + b − 2ab cos(θ) = c .
Apply this to the vector triangle of Figure 6.11 (lower), which has sides of length a = G ,
b = F , and c = G − F . Using property (6) of the dot product, we obtain
2 2 2
G + F −2 F G cos(θ) = G − F
2
2
= G + F −2G · F.
Assuming that neither F nor G is the zero vector, this gives us
F · G
cos(θ) = . (6.1)
F G
Since |cos(θ)|≤ 1 for all θ, equation (6.1) implies the Cauchy-Schwarz inequality:
|F · G|≤ F G .

b
c
θ
a

F
G – F
θ

G

FIGURE 6.11 The law
of cosines and the angle
between vectors.

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

October 14, 2010 14:21 THM/NEIL Page-155 27410_06_ch06_p145-186