Calculus: Analytic Geometry and Calculus, with Vectors

94 Vectors and geometry in three dimensions

This point P lies on ir if and only if D = 0, where

(2) D =

But

(3)

where

x1 + X(xs - x1) Y1 + X(ys - yl)

X3 Y3

x4 Y4

X6

(4) D, =

x1 y1 z, I

X3 y3 53 1

X4 Y4 Z4 1

X6 Y6 Z6 1

Y5

D = D1 + XD2,

D2 =

z,+X(zs-z,) 1+0

53 1

54 1

Z6 1

X2 - X1 Y2 - Y, Z2 - z1 0

X3 y3 53 1

X4

X5

Y4

Ys

Now D, 3 0 because P,, P3, P4, P6 do not lie in the same plane. In case

Ds = 0, there will be no X for which D = 0 and hence no point P on L which lies

in -r, so L and it must be parallel. In case Ds 5 0, there will be exactly one X for

which D = 0, that is, X = -D,/D3. Putting this in (1) then gives the coordi-

nates of the point of intersection of L and jr.

12 Find out whether the plane containing two vertices of a tetrahedron and

the mid-point of the opposite edge must contain the centroid of the tetrahedron.

13 This story can be read by anyone. It interests nearly everyone, but stu-

dents who do not have a lot of time at their disposal can postpone the pleasures

and benefits enjoyed by those who fully understand it. Matrices are often

denoted by single letters. For examples, w e ca n set

1 7 4 9

(1) P-

, Q-

1 1 1, R= -7 -^69

1 -1 1/ \0 1

1

-1

2 3

-4

S

6 -2

5

12

),

T

(

) U =

2

),

Y

=

8

-4 3 -2 3 \-1 \-4/

We shall, among other things, develop enough algebra of matrices to enable us

to understand and verify the fact that these matrices satisfy the conditions

PQ=R QP=S PQ0QP,

(2) QT = U, PU=Y, P(QT)=Y, (PQ)T=V

det (PQ) = det (QP) = (det P) (det Q).

If I is a square matrix, we can denote its determinant by det -4. Two matrices

A and B are said to be equal, and we write .4= B, if they (i) have the same num-

ber of rows, (ii) have the same number of columns, and (iii) have equal elements

in corresponding positions. We can multiply matrices by scalars (numbers),

and we can add two matrices which have thesame number of rows and the same

number of columns. For example, when P and Q are the matrices displayed