Name: ________________________ Class: ___________________ Date: __________
ID: A
M43 Test 3 Su2011
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
____
____
1. Find the vector v that has a magnitude of 6 and is in the same direction as u, where u = −6, −4 .
a.
v= −
3
2
,−
13
13
d.
v=
b.
v= −
2
3
,−
13
13
e.
v= −
c.
v= −
18
12
,−
13
13
4
,
13
2
,
13
6
13
2
13
2. Find the component form of v if Äv Ä = 8 and the angle it makes with the x-axis is 150°.
a.
−8 3, − 8
c.
−4 2, 4 2
b.
−8, 8 3
d.
−4 3, 4
e.
−4, 4 3
3. final If ÄuÄ = 3 and Äv Ä = 5, and the vectors make angles of 0° and 60° and with the x-axis
respectively, find the component form of the sum of u and v. Round answers to two decimal places.
____
a.
2.50, 7.33
c.
5.50, 4.33
b.
6.50, 2.60
d.
0.50, −4.33
e.
−2.50, −1.33
4. final Given that Force 1 = 35 pounds and Force 2 = 120 pounds, find the angle between the forces if
the magnitude of the resultant force is 140 pounds. Round answer to the nearest degree.
a. 118°
b. 123°
c. 110°
d. 115°
e. 130°
____
5. Given vectors u = 2, 1 and v = 1, −4 , determine the quantity indicated below.
(u ⋅ 4v)u
____
a.
−24, −12
c.
8, 4
b.
−28, −14
d.
−72, −36
e.
−16, −8
6. Use vectors to find the measure of the angle at vertex B of triangle ABC, when
A = (2, 4), B = (−2, 2), and C = (−3, −5) . Round answer to two decimal places.
123.51°
b. 122.61°
c. 124.70°
d. 126.04°
e. 127.22°
a.
1
Name: ________________________
____
7. Find the projection of u onto v if u = −3, 1 , v = −1, 5 .
a.
b.
____
ID: A
8
,
26
24
− ,
26
−
40
26
40
26
c.
d.
8
8
,
26 26
24
8
− ,−
26
26
−
e.
−
24 8
,
26 26
8. A 725-pound trailer is sitting on an exit ramp inclined at 36° on Highway 35. How much force is
required to keep the trailer from rolling back down the exit ramp? Round answer to two decimal
places.
a. 546.44 pounds
c. 426.14 pounds
e. 506.34 pounds
b. 566.49 pounds
d. 586.54 pounds
____
9. A force of 45 pounds is exerted along a rope attached to a crate at an angle of 30° above the
horizontal. The crate is moved 23 feet. How much work has been accomplished? Round answer to
one decimal place.
a. 896.3 foot-pounds
d. 1,035.0 foot-pounds
b. 965.7 foot-pounds
e. 1,080.4 foot-pounds
c. 1,195.1 foot-pounds
____ 10. Find the standard form of the equation of the sphere with the given characteristics.
Endpoints of a diameter: (–7, 4, 2), (9, 8, 4)
a.
2
(x − 2) 2 + ÁËÊ y − 12 ˜¯ˆ + ( z − 6) 2 = 69
d.
2
(x − 1) 2 + ÁËÊ y − 6 ˜¯ˆ + ( z − 3) 2 = 276
b.
2
(x + 1) 2 + ÁËÊ y + 6 ˆ˜¯ + ( z + 3) 2 = 69
e.
2
(x − 1) 2 + ÁËÊ y − 6 ˜¯ˆ + ( z − 3) 2 = 69
c.
(x − 1) 2 + ÁËÊ y − 6 ˆ˜¯ + ( z − 3) 2 = 138
d.
e.
center: (3, –5, –8); radius: 2
center: (–3, –5, –8); radius: 2
2
____ 11. Find the center and radius of the sphere.
2
2
2
x + y + z − 6x − 10y − 16z + 94 = 0
a.
b.
c.
center: (–3, –5, –8); radius: 4
center: (3, 5, 8); radius: 2
center: (–3, –5, –8); radius: 4
____ 12. Find the magnitude of the vector described below.
Initial point: (6, 6, 4)
Terminal point: (–9, –6, 9)
a. 22
b. 4 2
c.
54
d.
394
e.
8 2
____ 13. Find the angle between the vectors u and v. Express your answer in degrees and round to the nearest
tenth of a degree.
u = −5, −5, −9 , v = −8, 1, −2
a.
56.1°
b.
90°
c.
29.1°
2
d.
60.9°
e.
33.9°
Name: ________________________
ID: A
____ 14. Find the vector z, given u = 8, −7, −9 and v = −8, −2, 5 , and w = 56, 13, −80 .
3u – 5v + 4z = w
a.
2, −6, −7
c.
6, −2, −7
b.
−1, −6, −6
d.
−8, 24, −28
e.
−2, 6, −7
____ 15. Find a unit vector in the opposite direction of u.
u = −13, 7, 4
a.
1
−13, 7, 4
3 26
d.
b.
2 6 −13, 7, 4
e.
c.
1
13, −7, −4
3 26
1
13, −7, −4
2 6
13, −7, −4
____ 16. Determine whether u and v are parallel, orthogonal, or neither.
u = 8, −4, 7 , v = 40, −20, 35
a.
neither
b.
parallel
c.
orthogonal
____ 17. Use vectors to determine whether the points are collinear.
(9, –7, –6), (5, –9, –7), (13, –5, –5)
a.
collinear
b.
not collinear
____ 18. Find a unit vector orthogonal to u and v.
u = –3i – j + k, v = –3i + 2j + 3k
a.
b.
c.
1 Ê
Á −5i+ 6j− 9k ˆ˜
¯
142 Ë
1 Ê
Á 9i− 2j+ 3k ˆ˜
¯
10 Ë
9i− 2j+ 3k
1 Ê
Á 9i+ j+ k ˆ˜
¯
11 Ë
d.
e.
−5i + 6j− 9k
____ 19. Find the area of the triangle with the given vertices.
(5, –1, 2), (7,–4,–2), (2, –6, 3)
a.
b.
3 110
2
3 110
4
c.
0
d.
3 110
3
Name: ________________________
ID: A
____ 20. Find the triple scalar product u ⋅ (v × w) for the vectors
u = 5, 9, −6 , v = 5, 8, −1 , w = −2, 8, 3
a.
–59
b.
–293
c.
544
d.
293
e.
0
____ 21. Which oset of parametric equations represent the following line or conic.
Use x = h + a cos(θ) , y = k + b sin(θ):
Ellipse with vertices ÊÁË 6,5 ˆ˜¯ , ÊÁË 12,5 ˆ˜¯ and foci ÊÁË 7,5 ˆ˜¯ , ÊÁË 11,5 ˆ˜¯ .
a.
x = −5 +
5 cos(θ)
c.
x = 9+
5 cos(θ)
5 cos(θ)
y = −9 + 3sin(θ)
y = −9 + 3sin(θ)
b.
x = 5+
d.
y = 5 + 3sin(θ)
x = 9 + 3 cos(θ)
y = 5+
5 sin(θ)
____ 22. Which answer is a set of polar coordinates for the following rectangular coordinates. Answers are rounded to
3 decimal places.
ÁÊ −4,−2 ˜ˆ
Ë
¯
Ê
Á
a. Ë 4.472,1.107 ˜ˆ¯
c. ÁÊË −4.472,−0.464 ˜ˆ¯
b. ÊÁË −4.472,0.464 ˆ˜¯
d. ÊÁË 4.472,0.464 ˆ˜¯
____ 23. Which answer is a polar form of the given rectangular equation?
25xy = 225
a.
b.
r 2 = 9sec(θ) csc(θ)
r 2 = 9sin(θ) cos(θ)
c.
d.
r 2 = 3sec(θ) csc(θ)
r 2 = 3sin(θ) cos(θ)
____ 24. Find a polar equation of the conic with the given characteristics and with one focus at the pole:
Parabola with directrix x = 4.
4
4
a. r =
c. r =
1 − cos(θ)
1 + cos(θ)
1
4
b. r =
d. r =
1 + 4sin(θ)
1 + sin(θ)
____ 25. Find a polar equation of the conic with the given characteristics and with one focus at the pole:
ÊÁ 6 π ˆ˜ ÊÁ
3π ˆ˜˜˜
Hyperbola with vertices ÁÁÁÁ , ˜˜˜˜ , ÁÁÁÁ −6,
2 ˜˜¯
Ë5 2 ¯ Ë
a.
b.
6
2 − 3sin(θ)
6
r=
2 + 3sin(θ)
r=
c.
d.
4
6
2 + 3cos(θ)
6
r=
2 − 3cos(θ)
r=
ID: A
M43 Test 3 Su2011
Answer Section
MULTIPLE CHOICE
1. ANS:
OBJ:
2. ANS:
3. ANS:
4. ANS:
5. ANS:
6. ANS:
7. ANS:
8. ANS:
9. ANS:
10. ANS:
11. ANS:
12. ANS:
13. ANS:
14. ANS:
15. ANS:
16. ANS:
17. ANS:
18. ANS:
19. ANS:
20. ANS:
21. ANS:
22. ANS:
23. ANS:
24. ANS:
25. ANS:
C
PTS: 1
Determine vector given magnitude and direction of another vector
D
PTS: 1
C
PTS: 1
A
PTS: 1
E
PTS: 1
C
PTS: 1
A
PTS: 1
C
PTS: 1
A
PTS: 1
E
PTS: 1
OBJ: Find equation of sphere given center and radius
B
PTS: 1
OBJ: Find center and radius of sphere given equation
D
PTS: 1
OBJ: Find magnitude of vector in space
A
PTS: 1
OBJ: Find angle between vectors
E
PTS: 1
OBJ: Compute resultant vector
D
PTS: 1
OBJ: Find unit vector in space
B
PTS: 1
OBJ: Determine if vectors are orthogonal, parallel, or neither
A
PTS: 1
OBJ: Determine if points are collinear
A
PTS: 1
OBJ: Find unit vector for u x v
A
PTS: 1
OBJ: Find area of triangle defined vectorially
B
PTS: 1
OBJ: Find triple scalar product
D
PTS: 1
B
PTS: 1
A
PTS: 1
C
PTS: 1
B
PTS: 1
1