M ASSACHUSETTS I NSTITUTE OF T ECHNOLOGY
Interphase Calculus III Problem Set 2
Instructor: Samuel S. Watson
Due: 13 July 2015
1. Prove the following properties of determinants.


a b c
det  a b c  = 0
d e f



λ a λb λc
a
e
f  = λ det  d
det  d
g h
i
g



a b c
a
det  d e f  = − det  g
g h i
d
2. In class we learned that if L( x, y) =
a b
c d
x
y

c
f 
i

b c
h i 
e f
b
e
h
and K ( x, y) =
e f
g h
x
y
are linear
transformations from R2 to R2 , then
a b (i) det
represents the factor by which areas are multiplied under L (and similarly for
c d K), and
(ii) the matrix product
a b
c d
e f
g h
represents the composition L ◦ K.
a b
e f
a b
e f Show that det
= det c d det g h by direct calculation.
c d
g h
Explain in words why this should be the case, using facts (i) and (ii).
3. (#54 on page 802 in E&P) Show that the perpendicular distance D from the point P0 ( x0 , y0 , z0 )
to the plane ax + by + cz = d is
D=
| ax0 + by0 + cz0 − d|
√
.
a2 + b2 + c2
(Hint: Find a parametric equation describing the line that passes through P0 and is perpendicular
to the given plane. Find the value of the parameter corresponding to the point where the line
intersects the plane.)
4. (#59 on page 802 in E&P) The line L1 is described by the parametric equation ( x (t), y(t), z(t)) =
(3 + 2t, t, 4), and the line L2 passes through the points P(2, 1, −3) and Q(0, 8, 4). These two lines are
skew, meaning that they do not intersect and are not parallel. Find the shortest possible distance
between a point on L1 and a point on L2 . (Hint: use the result in #58, which says p
that the perpendicular distance from ax + by + cz − d1 = 0 to ax + by + cz − d2 = 0 is |d1 − d2 |/ a2 + b2 + c2 . If
you use this result, you should show that it is true.)
5. (based on #35 on page 814 in E&P) The parametric equations of a moving point are
x (t) = 3 cos 2t
y(t) = 3 sin 2t
z(t) = 8t.
(a) Sketch the path of the point from time t = 0 to time t = 1.
(b) Find the velocity, speed, and acceleration of the particle at time t = 7π/8.
6. (#55 on page 815 in E&P) A point moves with constant speed, so its velocity vector v satisfies
|v|2 = C.
Show that the velocity and acceleration vectors are always perpendicular to each other.
7. (#53 on page 837 in E&P) Prove that the projection onto the xy-plane of the intersection of the
plane x + y + z = 1 and the ellipsoid x2 + 4y2 + 4z2 = 4 is an ellipse.
8. (top left of page 844 in E&P) Sketch or describe the surface or solid described by the given
equations and/or inequalities.
(a) r = 3,
−1 ≤ z ≤ 1
(b) ρ = 2,
π/3 ≤ φ ≤ 2π/3
(c) 1 ≤ r ≤ 3,
−2 ≤ z ≤ 2
9. The parabola z = x2 , y = 0 is rotated around the z-axis. Write a cylindrical-coordinate
equation for the surface thereby generated.
10. (#43 on page 847 in E&P) The curvature κ of a curve in the plane given parametrically by
( x (t), y(t)) is defined by
| x 0 (t)y00 (t) − y0 (t) x 00 (t)|
κ=
.
( x 0 (t)2 + y0 (t)2 )3/2
Roughly speaking, κ quantifies how sharp of a turn the curve is making at each point in time.
Find the points on the curve y = sin( x ) where the curvature is maximal and the points where it is
minimal.