Calculus: Analytic Geometry and Calculus, with Vectors

92 Vectors and geometry in three dimensions

Now write formulas for the coefficients 11, B, C in the expansion

A=f1(x-x1)+B(y-yi)+C(z-zi).

Show that the graph of the equation A = 0 contains the three points Pi(xi,yi,zl),

P2(x2,y2,z2), P3(x3,y3,z3) Comment upon the result.
4 Let ITI denote the area of the triangle T having vertices at the points
P1, P2i P of Figure 2.591. With an eye on the figure, discover a way in which the
formula

(1) JTJ =y'

Z

y(x-xl)+y 2y2(x2-x) -y' 2

y2(x2

-xi)

can be obtained. After expanding the products, show that some of the terms

cancel out and that the formula can be

IY P(x,y) put in the form

P2(x2,Y2)

(2) (T! = ±.

xl

Figure 2.591

x X2 with the plus sign. It can be shown that

(2) is correct with the plus sign when the

vertices P, F1, P2 occur in positive (coun-

terclockwise) order, and that (2) is cor-

rect with the minus sign when the vertices P, Pb P2 occur in negative (clockwise)

order. The members of (2) are 0 when the points are collinear. Many people

remember this.

5 This and the next two problems, together with Problem 11 at the end of

the next section, show that if Y is the volume of the tetrahedron (or simplex)

in E3 having vertices (x,y,z), (xl,yl,z1), (x2,y2,z2), (xa,y3,z3), then

where the sign is chosen such that F > 0 (or Y >= 0 if we allow degenerate tetra-

hedrons to be called tetrahedrons). Verify that this formula is correct with the

negative sign for the special case in which a, b, c are positive constants and the

four vertices are, in order, (0,0,0), (a,0,0), (0,b,0), and (0,0,c). Hint: Remember

or learn that the volume of a simplex (tetrahedron) in E3 is one-third of the

product of the altitude (number, not line segment) and the area of the triangular

base.

(^6) Letting D be the determinant of the preceding problem, show that
D= -
X

-xl

-x2

-x,

y z 1

-yi -z1 -1

-y2 -Z= -1

-y3 -Z3 -1

x y z 1

x-XI y -yl z - Zl^0

x-X2 Y -y2 Z - Z2^0

x-x3 y - ys z -z3 0