2.2 Coordinate systems and vectors in Es 65which make the angle a with L. Supposing that 0 < a < a/2, sketch the cir-
cular cone whose vertex is the origin, whose axis is the z axis, and whose central
angle is a. Then show that the equation of this cone is z2 = k2(x2 + y2), where
k = cot a.
17 Sketch a figure, similar to Figure 2.27, in which the paraboloid opens to
the right along the y axis instead of upward along the z axis. Note that inter-
changing y and z in (2.28) gives the equation y = k(x2 + z2) of the new paraboloid.
18 Sketch a figure, similar to Figure 2.27, in which the paraboloid opens for-
ward along the x axis instead of upward along the z axis. Note that interchang-
ing x and z in (2.28) gives the equation x = k(y2 + z2) of the new paraboloid.
19 Plot the eight points (±2, ± 2, ± 2) obtained by taking all possible choices
of the plus and minus signs. Then connect these points by line segments to
obtain the edges of the cube of which the eight points are vertices. Remark:
One who finds this problem to be unexpectedly difficult need not be disturbed.
The problem is unexpectedly difficult.
20 We embark on a little excursion to learn more about our abilities to sketch
graphs. The graph in E2 of the equation xy = 1 does not intersect the coordinate
axes, and it consists of two parts (or branches) that are easily sketched. The
graph in Es of the equation xyz = 1 consists of those points in Es having coordi-
nates (x,y,z) for which xyz = 1. The graph does not intersect the coordinate
planes, and it consists of four parts, namely, the one containing some points for
which x > 0, y > 0, z > 0, the one containing some points for which x > 0,
y < 0, z < 0, the one containing some points for which x < 0, y > 0, z < 0,
and the one containing some points for which x < 0, y < 0, z > 0. Everyone
should discover for himself that it is surprisingly difficult (or hopelessly impossible)
to draw x, y, z axes on a flat sheet of paper and sketch a figure which shows
what these four parts look like and how they are situated relative to each other
and to the coordinate system.
21 Draw the rectangular coordinate system obtained from that in Figure
2.23 by interchanging the x and y axes and the i and j vectors. Work out the
formulas for the vector products of these vectors and show that the system is
left-handed. As a safety measure, make a note on your figure that it is left-
handed and be sure that your formulas for vector products are not remembered.
22 It is not necessarily true that our study of mathematical machinery is
made more difficult when we pause briefly
to look at a rather complicated appli-
cation of it. Figure 2.291 shows a circle
C in the yz plane which has its center at
the point (0,b,0) and has radius a. We
suppose that 0 < a < b. The surface
T obtained by rotating this circle C about
the z axis is called a torus. Thus a torus
is the surface of a ring or hoop that is 'b a
more or less closely approximated by an
automobile tire. As the figure indicates, Figure 2.291
each point P on the torus T lies on the
circle which (i) contains a point P' on C, (ii) lies in a plane parallel to the xy
plane, and (iii) has its center at a point Q on the z axis. When the angles 0 and