Calculus: Analytic Geometry and Calculus, with Vectors

74 Vectors and geometry in three dimensions

of checking the answer when

(a) u =2i-3j+4k,v=2i+3j+4k

(b) u = i + j, v = i

(c)u=i+j+k, vi

17 Show that the two vectors

u, = cos Bi + sin 9j

u2 = -sin Bi + cos Oj

constitute an orthonormal set, that is,

lull = 1, Iu2I = 1, and u,-u2 = 0-

18 Show that the three vectors

u, = cos sin Bi + sin i sin Bj + cos Ok

u2 = cos cos Si + sin 0 cos Bj - sin Bk

us = -sin Oi --cos Oj

constitute an orthonormal set, that is,

u,, u, is 1 when p = q and is 0 when p ,-E q.

19 This problem requires that we learn the procedure by which we obtain

a useful formula for the gravitational force F which is exerted upon a particle

P1 having mass ml and situated at the point Pi(x,,y,,z,) by another particle P2

having mass m2 and situated at the point P2(x2,y2,z2). The very modest Figure

2.394 can help us understand what we do. We start with the Newton law of

universal gravitation, which is an "intrinsic law" that does not depend upon coordi-

nate systems. This law says that there is a "universal constant" G, which

depends only upon the units used to measure force, distance, and mass, such that

a particle P1 of mass m, at P, is attracted toward a second particle P2 of mass

m2 at P2 by a force F having magnitude

P,

Figure 2.394

(1) IFl = G

m,m2

d2

'

where d is the distance between the points. Our next

step is to put (1) in the form

(2) IFI = G :

MIM2

IP7P212

According to the Newton law, the direction of F is the direction of the vector

P,P2 (notP2P,). We have learned (and can relearn if the fact has been forgotten)

that each nonzero vector F is the product of IFI, the length or magnitude of F,

and the unit vector F/IFI in the direction of F. When F has the direction of

P1P2, this unit vector is P1P2/(P1P2I. Therefore,

(3) F = G 'n1m2 P,P2 = Gm,-2 PIP2

IP1P'I2 IP1P21 IP,P218