Calculus: Analytic Geometry and Calculus, with Vectors

100 Vectors and geometry in three dimensions

hasty reading and preliminary ideas will be satisfactory preparation for
future encounters with the material.
Here we begin to explore some of the reasons why the system (2.63)

of equations is important. Suppose we have, as in Figure 2.64, two

z

op

Figure 2.64

right-handed rectangular coordinate systems in E3. The x, y, z coordi-

nate system having origin at 0 and bearing an orthonormal set i, j, k

of vectors is shown on the left. The x', y', z' coordinate system having

origin at 0' and bearing an orthonormal set i', j', k' of vectors is shown on

the right. Our first task is to study the important systems of equations

(2.65)

1' = aj i + a1gj + a13k,

j' = a2li -1- a22j + a23k,

k' = a31i + a32j + a33k,

aiii' + a2ij' + a31k'

j = a12i' + a22J' + a32k'

k = a13i' + a23j' + a33k'

that relate the vectors in the two orthonormal sets. We observe a fact

that can be considered to be remarkable even when we know the reason

for it: the coefficients in the system of equations obtained by solving

the first system for i, j, k are easily written down by interchanging the

rows and columns in the first system. The reason is simple. The

orthonormality of the vectors implies that, in each system, all is

a12 is an is a21 is and so on until, finally, a33 is k'-k. The

numerical coefficients in each row (and hence also in each column) in

the right member of each system are the scalar components of a unit

vector. The nine coefficients are cosines of direction angles, but we

carefully avoid attempts to work out formulas by means of figures

showing the nine angles.

As soon as we look at the point P of Figure 2.64, we realize that P has

two sets of coordinates, there being a set x, y, z for the unprimed or old

coordinate system and another set x', y', z' for the primed or new coordi-

nate system. If we know enough about the relative positions of the

two coordinate systems, we should be able to find one set of coordinates

when we know the other set. We shall solve this problem with the aid

of vectors. To specify the relation between the two coordinate systems,

we suppose that, with reference to the unprimed coordinate system, the