2.2 Coordinate systems and vectors in E3 63The final figure should look much like Figure 2.21. It seems that we do not
inherit abilities to do things like this neatly and correctly. A little practice is
needed, and it often happens that the first figures we draw are very clumsy.
4 In the xy and xz planes, sketch circles of radius 3 having their centers at
the origin. Then complete the sketch of the sphere of which these circles are
great circles, that is, intersections of the sphere and planes through the center of
the sphere.
5 A spherical ball of radius 3 has its center at the origin. Sketch the part
of it that lies in the first octant.
6 Sketch a rectangular x, y, z coordinate system and observe that, in each
case, the graph of the equation or system of equations on the left is the entity
(point set) on the right:
X = 0 yz plane
y = 0 xz plane
z = 0 xy plane
x = y = 0 z axis
x = z = 0 y axis
y = z = 0 x axis
x = y = z line through (0,0,0), (1,1,1)Remark: We make no effort to remember these facts, but whenever we see an
x, y, z coordinate system, we should be able to observe and use these facts as they
are needed.
7 There are many points P(x,y,z) whose coordinates satisfy the equation
y = 3. Some examples are (0,3,0), (0,3,1), (1,3,0), (1,3,1), and (-40,3,416).
Sketch a figure and become convinced that the graph of the equation y = 3 is
the plane ir which passes through the point (0,3,0) and is both perpendicular to
the y axis and parallel to the xz plane. Then, without so much attention to
details, describe the graph of the equation z = 2.
8 Plot the points (0,1,0) and (0,0,1) and then draw the line L through these
points. Show that if P(x,y,z) lies on L, then x = 0 and y + z = 1. Show also
that if x = 0 and y + z = 1, then P(x,y,z) lies on L. Remark: It is possible to
write a single equation equivalent to the system x = 0, y + z = 1. For exam-
ple, each of the equationsIxI+Iy+z-11 = 0
x2-I-(y+z-1)2 =0
does the trick. It is fashionable to keep the two equations, and one who wishes
to do so may learn something by thinking about the matter.
9 Put the equation x2 + y2 + z2 - 2x - 4y + 8z = 0 into the standard
form (2.26) of the equation of a sphere and find the center and radius of the sphere.
Hint: Complete squares. Check your result by observing that the coordinates
of the origin satisfy the given equation and hence that the distance from the
origin to the center of the sphere must be the radius of the sphere.
10 A set S consists of those points P in Es for which I API2 + IBPI2 = 16,
where A is the origin and B is the point (0,2,0). Show that S is the sphere of
radius having its center at the point (0,1,0). Sketch the coordinate system
and S.