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and whose magnitude is the area of the base parallelogram. From the defini-
r
u
tion of the scalar product, dotting this vector with will give a scalar that is
the product of the area of the parallelepiped base multiplied by the parallel-
epiped height, whose magnitude is exactly the volume of the parallelepiped.
The circular permutation property of Eq. (7.52) then has a very simple geo-
metric interpretation: in computing the volume of a parallelepiped, it does
not matter which surface we call base.

MATLAB Representation
r r
The cross product of the vectors u = ( u u u ) and , , v = ( v v v, , ) is found
1 2 3 1 2 3
using the cross(u,v) command. r r r
The triple scalar product of the vectors uv, ,and w is found through the
det([u;v;w]) command. Make sure that the vectors defined as arguments
of these functions are defined as 3-D vectors, so that the commands work and
the results make sense.

Example 7.8
r r r
Given the vectors = (2, 1, 0), = (0, 3, 0), u v w = (1, 2, 3), find the cross prod-
uct of the separate pairs of these vectors, and the volume of the parallelepi-
ped formed by the three vectors.

Solution: Type, execute, and interpret at each step, each of the following com-
mands, using the above definitions:

u=[2 1 0]
v=[0 3 0]
w=[1 2 3]
ucrossv=cross(u,v)
ucrossw=cross(u,w)
vcrossw=cross(v,w)
paralvol=abs(det([u;v;w]))
paralvol2=abs(cross(u,v)*w')
Question: Verify that the last command is an alternate way of writing the vol-
ume of the parallelepiped expression.

In-Class Exercises

Pb. 7.10 Compute the shortest distance from New York to London. (Hint:
(1) A great circle is the shortest path between two points on a sphere; (2) the
angle between the radial unit vectors passing through each of the cities can
be obtained from their respective latitude and longitude.)

© 2001 by CRC Press LLC