Practice Test 4
Sections 6.6, 6.7, 9.1, 9.2, and 9.3
Sketch the vector as a position vector and find its
magnitude.
1) v = -2i + 5j
12) v = 7, θ = 225°
Solve the problem.
13) The magnitude and direction of two forces
acting on an object are 35 pounds, N45°E, and
55 pounds, S30°E, respectively. Find the
magnitude, to the nearest hundredth of a
pound, and the direction angle, to the nearest
tenth of a degree, of the resultant force.
y
6
4
2
-6
-4
-2
2
4
6
x
-2
14) Two forces, F1 and F2 , of magnitude 60 and 70
-4
pounds, respectively, act on an object. The
direction of F1 is N40°E and the direction of F2
is N40°W. Find the magnitude and the
direction angle of the resultant force. Express
the direction angle to the nearest tenth of a
degree.
-6
Let v be the vector from initial point P1 to terminal point
P2 . Write v in terms of i and j.
Use the given vectors to find the specified scalar.
15) u = 11i + 4j and v = -6i - 5j; Find u · v.
2) P1 = (6, 4); P2 = (-5, -4)
3) P1 = (2, 3); P2 = (-6, 3)
16) v = 6i + 2j; Find v · v.
Find the specified vector or scalar.
4) u = -7i - 3j, v = -5i + 7j; Find u + v.
A) -13i + 4j
B) -2i - 13j
C) 2i + 4j
D) -12i + 4j
17) u = -6i + 10j, v = -7i - 6j; Find (-5u) · v.
18) u = 6i + 10j, v = -2i - 8j, w = -4i - 2j; Find u ·
(v + w).
5) u = -9i - 2j, v = 5i + 7j; Find u - v.
Find the angle between the given vectors. Round to the
nearest tenth of a degree.
19) u = -3i + 6j, v = 5i + 2j
6) v = 8i + 2j; Find 3v.
7) u = -7i + 1j and v = 8i + 1j; Find u + v .
20) u = 3j, v = 8i + 2j
Find the unit vector that has the same direction as the
vector v.
8) v = 5i + 12j
Use the dot product to determine whether the vectors are
parallel, orthogonal, or neither.
21) v = 3i + 2j, w = 3i - 2j
9) v = -9j
22) v = 3i - j, w = 6i - 2j
Write the vector v in terms of i and j whose magnitude v
and direction angle θ are given.
10) v = 8, θ = 30°
23) v = 4i + 3j, w = 3i - 4j
24) v = 3i, w = -2i
11) v = 12, θ = 180°
1
Solve the problem.
25) A person is pulling a freight cart with a force
of 59 pounds. How much work is done in
moving the cart 80 feet if the cart's handle
makes an angle of 30° with the ground?
Solve the problem.
31) The arch beneath a bridge is semi-elliptical, a
one-way roadway passes under the arch. The
width of the roadway is 38 feet and the height
of the arch over the center of the roadway is 11
feet. Two trucks plan to use this road. They are
both 8 feet wide. Truck 1 has an overall height
of 10 feet and Truck 2 has an overall height of
11 feet. Draw a rough sketch of the situation
and determine which of the trucks can pass
under the bridge.
Graph the ellipse and locate the foci.
x2 y2
26)
+
=1
25 49
y
10
Find the vertices and locate the foci for the hyperbola
whose equation is given.
32) 49x2 - 16y2 = 784
5
-10
-5
5
10
x
Match the equation to the graph.
y2 x2
33)
=1
9
16
-5
-10
A)
6
y
5
Graph the ellipse.
4
27) 16(x - 2)2 + 9(y - 1)2 = 144
3
y
2
10
1
5
-6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6x
1
2
3
4
5
6x
-2
-3
-10
-5
5
10
-4
x
-5
-6
-5
B)
-10
6
y
5
4
Find the standard form of the equation of the ellipse
satisfying the given conditions.
28) Major axis horizontal with length 12; length of
minor axis = 6; center (0, 0)
3
2
1
-6 -5 -4 -3 -2 -1
-1
-2
29) Foci: (0, -2), (0, 2); y-intercepts: -3 and 3
-3
-4
-5
Convert the equation to the standard form for an ellipse
by completing the square on x and y.
30) 4x2 + 16y2 + 8x - 64y + 4 = 0
-6
2
Convert the equation to the standard form for a hyperbola
by completing the square on x and y.
37) y2 - 16x2 - 2y - 32x - 31 = 0
C)
6
y
5
4
3
Find the location of the center, vertices, and foci for the
hyperbola described by the equation.
(x + 2)2 (y + 4)2
38)
=1
64
100
2
1
-6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6x
-2
-3
Find the standard form of the equation of the hyperbola.
39)
-4
-5
6
-6
y
5
D)
6
4
y
3
5
2
4
1
3
2
-6 -5 -4 -3 -2 -1
-1
1
-6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6x
-2
1
2
3
4
5
-3
6x
-4
-2
-5
-3
-6
-4
-5
A)
x2 y2
=1
9
25
B)
y2 x2
=1
25
9
C)
x2 y2
=1
25
9
D)
y2 x2
=1
9
25
-6
Find the standard form of the equation of the hyperbola
satisfying the given conditions.
34) Foci: (0, -10), (0, 10); vertices: (0, -6), (0, 6)
Use the center, vertices, and asymptotes to graph the
hyperbola.
(y - 2)2 (x + 1)2
40)
=1
4
16
35) Center: (6, 5); Focus: (3, 5); Vertex: (5, 5)
Use vertices and asymptotes to graph the hyperbola. Find
the equations of the asymptotes.
36) 9x2 - 4y2 = 36
y
10
y
5
10
-10
5
-5
5
10
x
-5
-10
-5
5
10
x
-10
-5
Find the focus and directrix of the parabola with the given
equation.
41) y2 = 28x
-10
3
Match the equation to the graph.
42) (x + 2)2 = 8(y + 2)
Find the standard form of the equation of the parabola
using the information given.
43) Focus: (0, 21); Directrix: y = -21
A)
10
y
Convert the equation to the standard form for a parabola
by completing the square on x or y as appropriate.
44) y2 - 4y - 2x - 2 = 0
5
-10
-5
5
10 x
Find the vertex, focus, and directrix of the parabola with
the given equation.
45) (x + 4)2 = 20(y + 2)
-5
-10
Graph the parabola with the given equation.
46) (y - 1)2 = 5(x + 2)
B)
10
y
10
y
5
5
-10
-5
5
10 x
-10
-5
-5
5
-5
-10
C)
-10
10
y
5
-10
-5
5
10 x
5
10 x
-5
-10
D)
10
y
5
-10
-5
-5
-10
4
10 x
Answer Key
Testname: PRACTICE TEST 4
1) v =
29
26) foci at (0, 2 6) and (0, -2 6)
y
y
10
6
4
5
2
-10
-6
-4
-2
2
4
6
-5
5
10
x
5
10
x
x
-5
-2
-4
-10
-6
27)
y
v = -11i - 8j
v = -8i
D
-14i - 9j
24i + 6j
5
5
12
8) u =
i+
j
13
13
2)
3)
4)
5)
6)
7)
10
5
-10
-5
-5
9) u = -j
10) v = 4 3i + 4j
11) v = -12i
7 2
7 2
12) v = ij
2
2
-10
13) F = 57.04; θ = -23.6°
14) F = 99.37; θ = 93.7°
15) -86
16) 40
17) 90
18) -136
19) 94.8°
20) 76°
21) neither
22) parallel
23) orthogonal
24) parallel
25) 4087.6 ft-lb
28)
x2 y2
+
=1
36
9
29)
x2 y2
+
=1
5
9
30)
(x + 1)2 (y - 2)2
+
=1
16
4
31) Truck 1 can pass under the bridge, but Truck 2
cannot.
32) vertices: (-4, 0), (4, 0)
foci: (- 65, 0), ( 65, 0)
33) D
y2 x2
34)
=1
36 64
(y - 5)2
35) (x - 6)2 =1
8
5
Answer Key
Testname: PRACTICE TEST 4
36) Asymptotes: y = ±
3
x
2
46)
10
y
y
10
5
5
-10
-10
-5
5
10
x
-5
5
-5
-5
-10
-10
37)
(y - 1)2
- (x + 1)2 = 1
16
38) Center: (-2, -4); Vertices: (-10, -4) and (6, -4); Foci:
(-2 - 2 41, -4) and (-2 + 2 41, -4)
39) B
40)
y
10
5
-10
-5
5
10
x
-5
-10
41) focus: (7, 0)
directrix: x = -7
42) D
43) x2 = 84y
44) (y - 2)2 = 2(x + 3)
45) vertex: (-4, -2)
focus: (-4, 3)
directrix: y = -7
6
10 x