Calculus: Analytic Geometry and Calculus, with Vectors

2.6 Vector products and changes of coordinates in E3 97

so that z = Ay, y = Bx, and z = A(Bx). Show that z = Cx, where the matrix
C is the product of 4 and B, that is, C = l1B. Remark: The result shows that
products of square matrices are defined in such a way that 11(Bx) = (.4B)x.
I5 A two-by-two matrix of numbers ak determines the system of equations

x' = affix + any

y' = a^_lx + ally

which transforms a given point (x,y) into its transform (x',v'). Let three points

(x1,yi), (x2>y2), (x3,y3) have transforms (xi,y ), (x2,Y2), (ta,Y3). As an exercise in

multiplying matrices (or determinants), prove that

x1

X2

X1

Yi 1

Y2 1

Y3 1

a11x1 + a12y1 a21x1 + I

a11x2 + a12y2 a21x2 + a22y2^1

a11x3 + a12y3 a21x3 + 1

X1

X2

X3

Y1^1

yz^1

Yz 1

all a21 0

a12 all 0

0 0 11

Remark: With the aid of the results of this problem, we can prove some theorems

in geometry. Let D = aua22 - a12a21, so that D is the determinant of the matrix

of the transformer. The area Ti 1of the triangular region having vertices at the

transforms is equal to IDI times the area I TI of the triangular region having ver-

tices at the original points. The orientation (clockwise or counterclockwise) is

preserved if D > 0 and is reversed if D < 0. If D = 0, the transforms are

collinear. The transformer conserves areas if and only if (DI = 1, that is,

D = 1 or D = -1. If the transformer is isometric (conserves distances), then

it also conserves areas, and hence IDS = 1. These results can be extended to

give information about transforms of oriented simplexes having four ordered

vertices (xk,yk,zk), k = 1, 2, 3, 4, in E3. When

D1 =

X1

xz

X3

D=

all a32 ala

a21 a22 a23

a31 a3: a33

the simplex is (by definition) positively oriented when D1 > 0 and negatively

oriented when D1 < 0. The transformer conserves volumes if and only if D = 1

or D = -1, and it preserves orientation if and only if D > 0.

2.6 Vector products and changes of coordinates in E3 Let u and v

be vectors in E3 having scalar components u1j u2, u3 and v1, V2, v3 with

respect to a right-handed x, y, z coordinate system endowed with the

usual unit vectors i, j, k. Then

(2.61) u = uji + uaj + u3k, v = vii + vzj + v3k.

The vector (or cross) product has been defined by the formula

(2.611) u X V = Jul lvf sin On