Calculus: Analytic Geometry and Calculus, with Vectors

S6 Vectors and geometry in three dimensions

17 Sketch a figure showing four points 0, P1i P2, and Q in Es and suppose
that IOQI = 1. Let Qi and Q2 be the projections of P1 and P2 on the line OQ.
Show that

(1) [ P OQIOQ = OQ1, [P1P2.OQ]OQ = Q Q2,

(2)

and hence that

[(OP1 + P1P2)'OQ]OQ = OQ2

(3) [(OP1 +P- P,.- =(OP-1 OQ + P1P2 OQ)OQ

and therefore

(4) (OP1 + P1P2)'OQ = OP1 OQ + P1P2 Q-

Remark: If we set u1 = OPI, U2 = P1P, and v = OQ, this shows that the formula

(5) (u1 +

is valid when u1 and u2 are vectors and v is a unit vector. It follows from this

that (5) is valid whenever u1, u2, and v are vectors. With the aid of (5) and

simpler properties of scalar products, we find that

(6) (u1 + v2) = (v1 + u2)

= u2) + u2)

= (u1 + (u1 +

and hence

(7) (u1 + v2) =u1v1 + u1v2 + u2v1 + u2v2.

This is the basic formula (2.183).

18 This problem and the next involve some very simple but very important

0

y ideas. Let r be the vector running from the

xi P(x,y) origin to P(x,y), the point P having coordi-

r nates x and y, as in Figure 2.198. Let i be a

unit vector having the direction of the posi-

tive x axis. Considering separately the cases

in which x > 0, x = 0, and x < 0, show that

i xi x xi is the vector running from the origin to the

Fignre 2.198 projection of P upon the x axis. Then let j

be a unit vector having the direction of the

positive y axis and prove that yj is the vector running from the origin to the

projection of P upon the y axis. Hint: All that is required is appropriate use of

Figure 2.199

the definition of the product of a scalar and a

vector.

19 As in Figure 2.199, let i and j be unit vectors

having the directions of the x and y axes of a plane

coordinate system. Let v be a nonzero vector

running from the origin to P(x,y). Show that

(1) v=xi+yj