Calculus: Analytic Geometry and Calculus, with Vectors

60 Vectors and geometry in three dimensions

at the origin. The intersection (or section) of the sphere and the yz plane
is a circle through the north and south poles which could be drawn with a
compass. The intersection of the sphere and the xy plane is the equa-
torial circle which appears in the flat figure to be a flattened circle. The
intersection of the sphere and the xz plane is a circle composed of two
meridians passing through the poles. The three coordinate planes, the
xy plane, the yz plane, and xz plane, cut E3 into eight parts called octants.
The octant containing points having only nonnegative coordinates is
called the first octant, and most people neither know nor care whether the

Figure 2.23

others are numbered.

We can learn about coordinate systems and, at

the same time, prepare ourselves to solve problems

of many types in mathematics and other sciences

by introducing vectors. As in Figure 2.23, let 1, j,

and k denote unit vectors (vectors of length 1) in

the directions of the positive x, y, and z axes.

Since these vectors are orthogonal (which means

that two different ones are orthogonal or perpen-

dicular), normalized (which means that each one

has unit length), and lie in E3 (Euclid space of three dimensions), we say

that they constitute an orthonormal set of vectors in E3. The definition

of scalar products given in (2.16) implies that

(2.231) 1, 1, I

and that u and v are two different ones of the vectors i, j,

and k. Similarly, the definition (2.17) of vector products implies that

(2.232) 1xi=0, jxj=0, k x k = 0

and that

(2.233) ixj=k, jxk=i, kxi=j

j x i = -k, k x j = -i, i x k= -j.

To help remember these formulas, we can notice that if we write the

ordered set

(2.234) 1, j, k, i, j, k

of vectors, then the vector product of two consecutive ones in this order is

the next but that changing the order of the factors changes the sign of the

product. A rectangular coordinate system is said to be right-handed

when the x, y, and z axes are so oriented (or arranged) that their ortho-

normal set i, j, k of vectors is such that the formulas (2.233) are correct;

otherwise, the system is left-handed. We shall use only right-handed

systems so that we can always use the formulas (2.233).