Math 215
Homework Set 2b: §§12.4 – 12.6
Fall 2020
The cross product has many important uses in mathematics. Among the more useful, particularly in this
class, is that the cross product of two vectors gives us an “oriented area.” Here is what we mean by
this. Suppose P is a parallelogram with sides a and b. The cross product a × b has both magnitude
and direction. The magnitude of a × b provides us with the area of P ; this you can verify (see book or
Exercise 17). The direction of a × b provides us with an orientation for P as follows. If we were to stand
on the parallelogram P , we might want to know if we were standing on the “top” or on the “bottom” of P
with respect to a × b. Since a × b is a vector that is perpendicular to both a and b, the cross product a × b
is either in the same direction as the vector from our feet to our head or it is in the opposite direction. If
it is in the same direction, then we are standing on “top” of P with respect to a × b, otherwise we are
standing on the “bottom” of P with respect to a × b.
Most of the following are modified versions of exercises from Stewart’s Multivariable Calculus.
25. (12.3, 12.4, and 12.5) Suppose ha, b, ci is a nonzero vector and P is the plane described by the
equation ax + by + cz + d = 0. Given two points (x0 , y0 , z0 ) and (x1 , y1 , z1 ) in three space,
how does one go about determining whether or not the two points lie on the same side of the plane.
Carefully explain your reasoning. [Advice: This is strongly, but not directly, related to the paragraph
at the top of this homework.]
26. (12.3 and 12.4) Trout are swimming up the Blackfoot River at 45 meters per hour (relative to an
observer on the bank of the river) to spawn, and their density is five fish per meter cubed. John,
and his sons Norman and Paul, love to fish along the Blackfoot River.14 The river runs parallel to
the x-axis, its surface is parallel to the plane z = 0, and the net they are using has a rectangular
opening that is half a meter wide by half a meter long with normal vector n. While they all put the
net into the river so that n · j = 0, each member of the family has their own special technique for
using the net. Advice: Draw pictures – this is a problem about visualizing what is happening, it is
not a problem about using math in clever ways.
(a) John puts the net into the river so that its opening is perpendicular to the surface of the water
(that is, n · k = 0), approximately how many fish does John catch in ten minutes?
(b) Norman puts the net into the river so that its opening is parallel to the surface of the water
(that is, n · k = ±.25 meters squared), approximately how many fish does Norman catch in
ten minutes?
(c) Paul puts the net into the river so that its opening is at an angle of thirty degrees to the surface
of the water, approximately how many fish does Paul catch in ten minutes?
27. (12.5) Obi-wan and Anakin are chasing Zam Wessel across the skylanes of the city of Coruscant.15
The ground is the plane z = 0 and Obi-wan and Anakin are at the point (325, 675, 550) while Zam
is at the point (776, 675, 599).
(a) Find parametric equations for the line of sight between the Jedi (Obi-wan and Anakin) and the
bounty hunter (Zam Wessel).
14
15
As in the move (and even better book) A River Runs Through It
As in the movie Star Wars: Attack of the Clones
(b) If we divide the line of sight into five equal segments, the heights of the buildings at the four
intermediate points from Obi-wan and Anakin to Zam Wessel are 559, 566, 576, and 589.
Ignoring other buildings, can Zam see Obi-wan and Anakin?
MatLab
(a) Graph the curve with parametric equations
~r(t) = h
27
8
−27
8
144
sin(8t) −
sin(18t),
cos(8t) +
cos(18t),
sin(5t)i.
26
39
26
39
65
(b) Show that the curve lies on the hyperboloid of one sheet 144x2 + 144y 2 − 25z 2 = 100.