Calculus: Analytic Geometry and Calculus, with Vectors

2.5 Determinants and applications 93

and hence that the formula for the volume of the tetrahedron can be put in the

form

Y= ±g

y - yi z- zi

Y -Y2 z-z2

Y-333 z-z3

7 Three vectors u and v and w with scalar components ul, u2, u3 and v1, v2, v3

and wi, w2, w3 have their tails at the same point P and are edges of a tetrahedron

having volume Y. Show that the formulas for F in the two preceding problems

can be put in the form

Y=±g

u1 U2 u3

Vi V2^93

W1 w2 W3

8 Supposing that Pi(xi,yi), P2(x2,y2), and Pa(x3,y3) are three noncollinear

(not on a line) points in E2, show that the equation

=0

has the form

11(x2+y2)+Bx+Cy+D=0

and that the graph contains the three points. Comment upon this result.

9 Show that if Pi(xi,yi), .. , Ps(x5,y5) are five different points in

then the equation

x2

XI^2

2

X2 2

X3

X4^2

X5^2

has the form

x33

xiyi

x2Y2

x3333

x4334

x5Y5

Y2 2

Yi

Y2 2

Y3

Y4

Ys

X

xi

X2

X3

X4

X5

=0

E2,

11x2+Bxy+Cy2+Dx+Ey +F=0

and that its graph contains the five points.

10 Use the ideas of the above examples to obtain the equation of asphere in

E3 which contains (or passes through) an appropriate collection of givenpoints.

11 Let Pl(xi,yi,zi), , P5(x5,ys,z5)be five given points (the coordinates

are supposed to be known numbers) such that nofour of them lie in the same

plane. Tell how to decide whether the line L containing Pi, P2 is parallel tothe

plane a containing P3, P4, Ps and tell how to find the point of intersectionwhen

L intersects ir. Solution: There are many ways to attack this problem. The

following method gives answers in terms of the known coordinates withrelatively

little calculation. A point P(x,y,z) lies on the line L if and only if, for some con-

stant X,

(1) x = xi + X (X9 - xi), 7 = yi + a(v2 - yi), % = z3 + X(z2 -- zi).