Name
ID
MATH 311
Exam 1
Spring 2001
Section 200
Solutions
P. Yasskin
1
/15 5
/10
2
/30 6
/10
3
/5 7
/10
4
/10 8
/10
1. (15 points) Consider the three points
P Ÿ1, 2, 3 Q Ÿ1, 3, 4 R Ÿ2, 3, 3 a. (5 pts) Find parametric equations of the plane containing P, Q and R.
PQ Q " P Ÿ0, 1, 1 x
y
PR R " P Ÿ1, 1, 0 X P sPQ tPR
0
1t
1
z
s
2
1
t
1
1
3
1
2st
3s
0
b. (5 pts) Find a non-parametric equation of the plane containing P, Q and R.
PQ • PR N
i
j
k
i Ÿ"1 " j Ÿ"1 k Ÿ"1 Ÿ"1, 1, "1 0 1 1
1 1 0
X N
P
N
" x y " z "1 2 " 3 "2
c. (5 pts) Find an equation of the line through P perpendicular to the plane containing P, Q and R.
X P tN
x
y
z
2
3
1"t
"1
1
t
1
"1
2t
3"t
1
2. (30 points) Consider the equations
2x 4y 6z 4
vy 2
wz b
2v w x 4y 4z 3
a. (5 pts) Write out the augmented matrix for this system.
0 0 2 4 6
4
1 0 0 1 0
2
0 1 0 0 1
b
2 1 1 4 4
3
b. (10 pts) For what value(s) of b do there exist solutions?
1 R and reorder rows :
2 1
R4
" 2R 1 " R 2 " R 3
¯
R4 :
1 0 0 1 0
2
0 1 0 0 1
b
0 0 1 2 3
2
2 1 1 4 4
3
1 0 0 1 0
2
0 1 0 0 1
b
0 0 1 2 3
2
0 0 0 0 0
"3 " b
So there are solutions only if b "3.
c. (10 pts) For those value(s) of b, find all solutions.
For b "3, the equations reduce to
vy 2
w z "3
x 2y 3z 2
So y and z are arbitrary and the solution is
2"s
v
w
x
"3 " t
2 " 2s " 3t
"1
2
"3
2
0
0
s
"2
t
"1
"3
y
s
0
1
0
z
t
0
0
1
d. (5 pts) Circle the geometric description of the solution set:
point,
line,
2-plane ,
3-plane,
4-plane,
R5
2
3. (5 points) In R 5 with coordinates Ÿv, w, x, y, z , write out an equation of the 3-plane through the point
P Ÿ5, 4, 3, 2, 1 with tangent vectors
a Ÿ2, 1, 3, 0, 4 b Ÿ1, "1, 2, "2, 3 v
5
2
1
2
w
4
1
"1
1
x
3
r
s
3
t
2
c Ÿ2, 1, 3, "1, 0 3
y
2
0
"2
"1
z
1
4
3
0
4. (10 points) Duke Skywater is flying the Millennium Eagle along the curve
rŸt Ÿ2 cos t, 3 sin t, t At t = , he releases a garbage pod which travels along his tangent line with constant velocity (equal
2
to his velocity at the time of release). Where is the garbage pod at t =?
rŸt Ÿ2 cos t, 3 sin t, t r =
vŸt Ÿ"2 sin t, 3 cos t, 1 v =
2 cos = , 3 sin = , = 0, 3, =
2
2 2
2
=
=
"2 sin , 3 cos , 1 Ÿ"2, 0, 1 2
2
The tangent line is
r tan Ÿt r = v =
Ÿx, y, z 2
2
At t =, the garbage pod is at
r tan Ÿ= 0, 3, = =
Ÿx, y, z 2
2
2
2
t" =
2
Ÿ
0, 3, =
2
t" =
2
Ÿ
"2, 0, 1 "2, 0, 1 Ÿ"=, 3, = 3
5. (10 points) Duke Skywater is flying the Millennium Eagle through the galactic polaron field. At t 20,
Duke’s position is r Ÿ20, 10, 30 lightyears and his velocity is v Ÿ.1, .3, .2 lightyears
year . At that time, he
polarons
and the gradient of this density to be
measures the density of polarons to be > 15 • 10 6
3
> Ÿ2 • 10 6 , "1 • 10 6 , 3 • 10 6 polarons
lightyear 4
lightyear
.
a. What does he measure as the time rate of change the polaron density,
d>
?
dt
d>
> v Ÿ2 • 10 6 , "1 • 10 6 , 3 • 10 6 Ÿ.1, .3, .2 Ÿ.2 " .3 .6 • 10 6 .5 • 10 6 5 • 10 5
dt
b. Using a linear approximation, what would he expect the polaron density to be at the point
x Ÿ21, 12, 29 ?
> x " r
> tan Ÿx > r > tan Ÿ21, 12, 29 15 • 10 6 Ÿ2 • 10 6 , "1 • 10 6 , 3 • 10 6 Ÿ1, 2, "1 15 • 10 6 Ÿ2 " 2 " 3 • 10 6 12 • 10 6
6. (10 points) Duke Skywater is flying the Millennium Eagle through the galactic hyperon field. At t 20,
Duke’s position is r Ÿ20, 10, 30 lightyears and his velocity is v Ÿ.1, .3, .2 lightyears
year . At that time, he
measures the hyperon field and its Jacobian to be
200
H
150
30
DH
Hans
"10 20
"40 10 5
"10 0 10
300
Hans
lightyear
a. What does he measure as the time rate of change the hyperon field, dH ?
dt
30
dH
v DH
dt
"10 20
3"34
.1
"40 10 5
"10 0 10
.3
.2
4
"4 3 1
"1 0 2
0
1
b. Using a linear approximation, what would he expect the hyperon field to be at t 22?
Ÿa dH
tan Ÿt H
H
Ÿt " a dt
200
4
tan Ÿ22 H
150
0
300
208
Ÿ
22 " 20 1
150
302
r tan Ÿ22 r tan Ÿ20 vŸ20 Ÿ22 " 20 Ÿ20, 10, 30 2Ÿ.1, .3, .2 Ÿ20.2, 10.6, 30.4 OR
rŸa DH
rŸt " rŸa H tan r tan Ÿt H
tan r tan Ÿ22 H
200
150
300
30
"10 20
"40 10 5
"10 0 10
.2
.6
.4
208
150
302
4
7. (10 points) Consider the vector space R of all positive real numbers with the operations of
Vector Addition:
x ¥ y xy
(real number addition)
)
Scalar Multiplication:
)©x x
(real number exponentiation)
Translate each of the following statements into ordinary arithmetic.
a. For all x we have 0 © x 0.
For all x we have x 0 1.
0.
b. For all a we have a © 0
For all a we have 1 a 1.
c. If a © x 0 then either a 0 or x 0.
If x a 1 then either a 0 or x 1.
8. (10 points) Consider the linear function L : R 3
¯
R 2 given by
1
; Ÿu 1 u 2 x u 3 x 2 dx
LŸ
u 0
d Ÿu u 2 x u 3 x 2 dx 1
x1
Find the matrix A of the linear function, so that you can rewrite it as
LŸ
u A u
1
1
L
L
L
0
; Ÿ1 dx
0
d Ÿ1 dx
0
; Ÿx dx
1
1
1
0
x1
0
0
d Ÿx dx
0
; Ÿx 2 dx
0
0
1
2
1
x1
1
d Ÿx 2 dx
1
A
0
1
2
1
1
3
2
1
3
2
0
1
x1
OR
1
LŸ
u ; Ÿu 1 u 2 x u 3 x 2 dx
0
d Ÿu u 2 x u 3 x 2 dx 1
u1 u2 1 u3 1
2
3
u 2 2u 3
2
3
u1x u2 x u3 x
2
3
Ÿu 2 2u 3 x | x1
x1
1
0
1
2
1
1
3
2
u1
u2
u3
A
1
0
1
0
1
2
1
1
3
2
5