Calculus: Analytic Geometry and Calculus, with Vectors

2.1 Vectors in E, 53

Problems 2.19

1 As in Figure 2.191, let A, B, C, , H be equally spaced points on the
line P1P2 with C = PI and G = P2. Apply appropriate definitions of the text
to show that

PD=P1P2, PE=P1P2, P1F=APP, PIG=PIP2

P H = IP P , P1B = -TP;P2, F1-4 = -3P1P2, DE=xPIP2.

Observe that the vector PIP lies on the line P1P2 if and only if there is a scalar
(or number or constant) X (lambda) such that

PIP = XP1P2.

Observe that the points P1 and P2 separate the line into three parts and tell what
values of X correspond to points in the different parts.
2 Construct a figure similar to Figure 2.191 which
shows points PI and P2 and also points .4, B, C, D for
which

P1:4 = -gP P, PIB= 8P1P21

PIC = $PIP, PID= I P2-

3 Let 0 (an origin), PI, and P2 be three points in E3
with PI -7- P2 as in Figure 2.192. Verify that if P is a
point on the line P1P2, then there is a scalar X for which

and

so

P1P = XP1P2 = X (OP2 -

01P1)

OP = PI + PIP = OPI + A(PP2 - OPI)

OP = XOP2 + (1 - X)OP1.

Show that if M is the mid-point of the line segment PIP2,

then

PM = (PPI + OP2).

4 Let i, j, and k be mutually perpendicular vectors

which run along bottom and back edges of a cube as in

Figure 2.193. Let PI, P2, P3, P4 be the mid-points of the

top edges upon which they lie. Show that

OPI=yi+k, OP2=i+-'ffj+k,

P4

Figure 2.193

and write similar formulas for U A, 0P41 and QP2.

(^5) Supposing that the vectors i, j, k of the preceding problem are unit vectors,
apply the definitions of products of vectors to prove that
1Xi=O, 1Xj=k.
Hint: In each case, write the definition of the product and use the angle correctly.
Figure 2.191
PI, k
P3