Page 180 - Advanced engineering mathematics

160 CHAPTER 6 Vectors and Vector Spaces

having the standard unit vectors in the first row, the components of F in the second row, and the
components of G in the third row. If this determinant is expanded by the first row, we get exactly
F × G:

i j k

a 1 b 1 c 1

a 2 b 2 c 2

b 1 c 1 a 1 c 1 a 1 b 1

= i − j + k
b 2 c 2 a 2 c 2 a 2 b 2

= (b 1 c 2 − b 2 c 1 )i + (a 2 c 1 − a 1 c 2 )j + (a 1 b 2 − a 2 b 1 )k
= F × G.
The cross product of two 3-vectors can be computed in MAPLE using the CrossProduct
command, which is part of the VectorCalculus package.
We will summarize some properties of the cross product.

1. F × G =−G × F.
2. F × G is orthogonal to both F and G. This is shown in Figure 6.14.
3. F × G = F G sin(θ) in which θ is the angle between F and G.
4. If F and G are nonzero vectors, then F × G = O if and only if F and G are parallel.
5. F × (G + H) = F × G + F × H.
6. α(F × G) = (αF) × G = F × (αG).

Property (1) of the cross product follows from the fact that interchanging two rows of a
determinant changes its sign. In computing F × G, the components of F are in the second row
of the determinant, and those of G in the third row. These rows are interchanged in computing
G × F.
For property (2), compute the dot product
F · (F × G)

= a 1 [b 1 c 2 − b 2 c 1 ]+ b 1 [a 2 c 1 − a 1 c 2 ]+ c 1 [a 1 b 2 − a 2 b 1 ]= 0.
Therefore, F is orthogonal to F × G. Similarly, G is orthogonal to F × G.

F × G
G

F
FIGURE 6.14 F×G is orthogonal
to F and to G.

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October 14, 2010 14:21 THM/NEIL Page-160 27410_06_ch06_p145-186