Exercicios resolvidos James Stewart vol. 2 7ª ed - ingles

!QP2\ = J [x2 - ~(X 1 + X2)] 2 + [y2- ~(Y l + Y2)] 2 + [z2 - ~ (z1 + z2)) 2
SECTION 12.1 THREE-DIMENSIONAL COORDINATE SYSTEMS 0 113
= (~x2- ~ x1) 2 + ( h2- ~ ·yt) 2 + (tz2 - ~z1 ) 2 = J G) 2 [(x2- xd + (y2 - yl) 2 + (z2 - z1) 2 )
.= ~J(x2 - xd + (y2- y1) 2 + (z2- zd = ~ \P1P2\
So Q is indeed the midpoint of P 1P2.
(b) By part (a), the midpoints of sides AB, B C and C A are P1 ( -~, 1, 4), P2 (1, ~. 5) and P3 (~ , ~, 4). (Re ~all that a median
of a triangle is a line segment from a vertex to the midpoint of the opposite side.) Then the lengths of the medians are:
IAP2\ = Jo 2 + (~ - 2) 2 + (5 - 3) 2 = jf+4 = -fj = ~
\BP3\ = J (~ + 2) 2 + (~) 2 + (4- 5) 2 = J¥ + * + 1 = J2j = ~JW
\CP1\ = J (-~- 4) 2 + (1 :__ 1} 2 + (4- 5? = J¥ + 1 = ~.J85
21 . (a) Since the sphere touches the xy-plan.e, its radius is the distance from its center, (2, -3, 6), to the xy-plane, namely 6.
Therefore r = 6 and an equation of the sphere is (x- 2? + (y + 3? + (z - 6? = 6 2 = 36.
(b) The radius of this sphere is the distance from its center (2, - 3, 6) to the yz-plane, which is 2. Therefore, an equation is
(~- 2) 2 + (y + 3) 2 + (z - 6) 2 = 4.
(c) Here the radius is the distance from the center (2, - 3, G) to the xz-plane, which is 3. Therefore, an equation is
(x- 2) 2 + (y + 3) 2 + (z- 6} 2 = 9.
23. The equation x = 5 represents a plane parallel to the yz-plane and 5 units in front of it.
25. The inequality y < 8 represents a half-space consisting of all points to the left of the plane y = 8.
27. The inequality 0 ~ z ~ 6 represents all points on or between the horizontal planes z = 0 (the xy-plane) and z = 6.
29. Because z = - 1, all points in the region must lie in the horizontal plane ~ = -1. In addition, x 2 + y 2 = 4, so the region
consists of all points that lie on a circle with radius 2 and center on the z-axis that is contained in the plane z = -1.
31. The inequality x 2 + y 2 + z 2 ~ 3 is equivalent to jx 2 + y 2 + z 2 ~ J3, so the region consists of those points whose distance
from the origin is at most V3. This is the set of all points on or inside the sphere with radius ../3 and center (0, 0 , 0).
33. Here x 2 + z 2 ~ 9 or equiva lently Jx2 + z 2 ~ 3 which describes the set of all points in JR 3 whose distance from they-axis is
at most 3. Thus, the inequality represents the region consisting of all points on or inside a circular cylinder of radius 3 with
axis the y-axis.
35. This describes all points whose x-coordinate is between 0 and 5, that is, 0 < x < 5.
37. This describes a region all of whose points have a distance to the origin which is greater than r, but smaller than R. So
inequalities describing the region are r < jx 2 + y 2 + z 2 < R, or r 2 < x 2 -1- y 2 + z 2 < R 2 .
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