Calculus: Analytic Geometry and Calculus, with Vectors

2.6 Vector products and changes of coordinates in Ea 109

which says that Q1 is the point on the line P3P4 which lies in the plane72 con-
taining P5, Pe, Q. Thus the required coordinates of Q, Q1, Q2are determined in
terms of X. Remark: Study of the set S of points P(x,y,z) that lieupon trans-
versals can be very interesting. If P lies upon the transversal through Q, then,
for some scalar μ,

(4) OP = (1 -μ)0Q + μ0Q1

But

(5) OQ = (1 - X)OP1 + AOP2,

and, since (3) shows that there are constants .4 and B for shich h1 = !A + B,

(6) OQ1= (1 - [sly + B])OP3 + (AX + B)OP4.

Therefore,

(7) OP = (1 - μ)(1 - )OP1 + (1 - μ)XOP2 + μ0P3

• μ(AX + B)OP3 + μ(_A +B)OP4.
  Hence there are vectors v1, v2, v3, v4 such that

(8) OP= V1 + Xv2 +μV3 + aμv4

It follows that there are scalars, a1, , d3 such that

x=ai+b1X+r1μ+d1Xμ

(9) y = a2 + b2X + e2μ + d2Xμ

Ia = a3 + b3X + c3t1 + d3aμ

It can be shown that the equations (9) are parametric equations of a quadric

surface. In fact, eliminating X and μ from the equations (9) shows that x, y, z

must satisfy an equation of the form (2.68). Thus S is a quadric surface, and

we have a quite straightforward procedure for determining its equation in terms

of the eighteen given coordinates of the six given points P1, P2, - - , P6. Stu-

dents who attain full comprehension of this matter will have passed far beyond

the minimum requirements of this course, and they can find the experience to

be both enjoyable and beneficial.

18 Those who wish to extend acquaintance with matrix theory should copy

the systems of equations in (2.65) and look at them while reading this. Let U

and UT denote the matrices of the coefficients (or scalar components or direction

cosines) of the systems so that

U

= (all a12

a2l a22 a2 a12 4

22

a

/' UT

=

\a31 all a33 a13 a23 a333

The matrix UT is called the transpose (or transposed matrix) of the matrix U,

and this invites us to realize that UT can be obtained from U by transposing

(interchanging) the rows and columns of U or by transposing the elements of U

across its main diagonal. The rows of U are scalar components of orthonormal

vectors, and the matrix is square. Such matrices are called unitary (or ortho-

normal) matrices. Therefore, U is unitary. When U is unitary, an application

of the rule for multiplying matrices shows that UUT = 1, where I is the unit