MA112 Section 750001: Prepared by Dr. Archara Pacheenburawana
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Exercise
Exercise 1.1
1 − 8 Find the vertex, focus, and directrix of the parabola and sketch its graph.
1. x = 2y 2
2. 4y + x2 = 0
3. 4x2 = −y
4. y 2 = 12x
5. (x + 2)2 = 8(y − 3)
6. x − 1 = (y + 5)2
7. y 2 + 2y + 12x + 25 = 0
8. y + 12x − 2x2 = 16
9 − 10 Find an equation of the parabola. Then find the focus and directrix.
9.
10.
y
y
1
1
−2
x
2
x
11 − 16 Find an equation for the parabola that satisfies the given conditions.
11. Vertex (0, 0), focus (0, −2)
12. Vertex (0, 0), directrix x = −5
13. Focus (−4, 0), directrix x = 2
14. Focus (3, 6), vertex (3, 2)
15. Vertex (0, 0), axis y = 0, passing through (1, −4)
16. Vertical axis y = 0, passing through (−2, 3), (0, 3), and (1, 9)
17 − 22 Find the vertices and foci of the ellipse and sketch its graph.
17.
x2 y 2
+
=1
9
5
18.
x2
y2
+
=1
64 100
19. 4x2 + y 2 = 16
20. 4x2 + 25y 2 = 25
21. 9x2 − 18x + 4y 2 = 27
22. x2 + 2y 2 − 6x + 4y + 7
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23 − 24 Find an equation of the ellipse. Then find its foci.
23.
24.
y
y
1
1
0
1
x
2
x
25 − 32 Find an equation for the ellipse that satisfies the given conditions.
25. Foci (±2, 0), vertices (±5, 0)
26. Foci (0, ±5), vertices (0, ±13)
27. Foci (0, 2), (0, 6), vertices (0, 0), (0, 8)
28. Foci (0, −1), (8, −1), vertex (9, −1)
29. Center (2, 2), focus (0, 2), vertex (5, 2)
30. Foci (±2, 0), passing through (2, 1)
31. Ends of major axis (0, ±6), pass through (−3, 2)
32. Foci (−1, 1) and (2, −3), minor axis of length 4
33 − 38 Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph.
33.
x2
y2
−
=1
144 25
34.
y2
x2
−
=1
16 36
35. y 2 − x2 = 4
36. 9x2 − 4y 2 = 36
37. 2y 2 − 3x2 − 4y + 12x + 8 = 0
38. 16x2 − 9y 2 + 64x − 90y = 305
39 − 46 Find an equation for a hyperbola that satisfies the given conditions.
39. Foci (0, ±3), vertices (0, ±1)
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40. Foci (±6, 0), vertices (±4, 0)
41. Foci (1, 3) and (7, 3), vertices (2, 3) and (7, 3)
42. Foci (2, −2) and (2, 8), vertex (2, 0) and (2, 6)
43. Vertices (±3, 0), asymptotes y = ±2x
44. Foci (2, 2) and (6, 2), asymptotes y = x − 2 and y = 6 − x
45. Vertices (0, 6) and (6, 6), foci 10 units apart
46. Asymptotes y = x − 2 and y = −x + 4, pass through the origin
47 − 52 Identify the type of conic section whose equation is given and find the
vertices and foci.
47. x2 = y + 1
48. x2 = y 2 + 1
49. x2 = 4y − 2y 2
50. y 2 − 8y = 6x − 16
51. y 2 + 2y = 4x2 + 3
52. 4x2 + 4x + y 2 = 0
Exercise 1.2
1. Let an x0 y 0-coordinate system be obtained by rotating an xy-coordinate system through
an angle of θ = 60◦ .
(a) Find the x0 y 0-coordinates of the point whose xy-coordinates are (−2, 6).
√
(b) Find an equation of the curve 3xy + y 2 = 6 in x0 y 0 -coordinates
(c) Sketch the curve in part (b), showing both xy-axes and x0 y 0-axes.
2 − 6 Rotate the coordinate axes to remove the xy-term. Then identify the type of
conic and sketch its graph.
2. xy = −9
3. x2 + 4xy − 2y 2 − 6 = 0
√
√
4. x2 + 2 3xy + 3y 2 + 2 3x − 2y = 0
5. 9x2 − 24xy + 16y 2 − 80x − 60y + 100 = 0
6. 52x2 − 72xy + 73y 2 + 40x + 30y − 75 = 0
7. Let an x0 y 0-coordinate system be obtained by rotating an xy-coordinate system through
an angle of θ = 45◦ . Find an equation of the curve 3x02 + y 02 = 6 in xy-coordinates.
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8 − 9 Show that the graph of the given equation is a parabola. Find its vertex,
focus, and directrix.
√
√
8. x2 + 2xy + y 2 + 4 2x − 4 2y = 0
9. 9x2 − 24xy + 16y 2 − 80x − 60y + 100 = 0
10 − 11 Show that the graph of the given equation is an ellipse. Find its foci,
vertices, and the ends of its minor axis.
10. 288x2 − 168xy + 337y 2 − 3600 = 0
√
√
11. 31x2 + 10 3xy + 21y 2 − 32x + 32 3y − 80 = 0
12 − 13 Show that the graph of the given equation is a hyperbola. Find its foci,
vertices, and asymptotes.
√
12. x2 − 10 3xy + 11y 2 + 64 = 0
√
√
13. 32y 2 − 52xy − 7x2 + 72 5x − 144 5y + 900 = 0
Exercise 2.1
1. Find the slope of the tangent line to the parametric curve x = t/2, y = t2 + 1 at
t = −1 and at t = 1 without eliminating the parameter.
2. Find the slope of the tangent line to the parametric curve x = 3 cos t, y = 4 sin t at
t = π/4 and at t = 7π/4 without eliminating the parameter.
3 − 8 Find dy/dx and d2 y/dx2 at the given point without eliminating the parameter.
3. x =
√
t, y = 2t + 4; t = 1
4. x = 12 t2 + 1, y =
1 3
t
3
− t; t = 2
5. x = sec t, y = tan t; t = π/3
6. x = sinh t, y = cosh t; t = 0
7. x = θ + cos θ, y = 1 + sin θ; θ = π/6
8. x = cos φ, y = 3 sin φ; φ = 5π/6
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9. (a) Find the equation of the tangent line to the curve
x = et ,
y = e−t
at t = 1 without eliminating the parameter.
(b) Find the equation of the tangent line in part (a) by eliminating the parameter.
10. (a) Find the equation of the tangent line to the curve
x = 2t + 4,
y = 8t2 − 2t + 4
at t = 1 without eliminating the parameter.
(b) Find the equation of the tangent line in part (a) by eliminating the parameter.
11 − 12 Find all values of t at which the parametric curve has (a) a horizontal tangent
line and (b) a vertical tangent line.
11. x = 2 sin t, y = 4 cos t
(0 ≤ t ≤ 2π)
12. x = 2t3 − 15t2 + 24t + 7, y = t2 + t + 1
Exercise 2.2
1 − 2 Plot the points in polar coordinates.
1. (a) (3, π/4)
(d) (4, 7π/6)
2. (a) (2, −π/3)
(d) (−5, −π/6)
(b) (5, 2π/3)
(c) (1, π/2)
(e) (−6, −π)
(f) (−1, 9π/4)
(b) (3/2, −7π/4)
(c) (−3, 3π/2)
(e) (2, 4π/3)
(f) (0, π)
3 − 4 Find the rectangular coordinates of the points whose polar coordinates are
given.
3. (a) (6, π/6)
(d) (0, −π)
4. (a) (−2, π/4)
(d) (3, 0)
(b) (7, 2π/3)
(c) (−6, −5π/6)
(e) (7, 17π/6)
(f) (−5, 0)
(b) (6, −π/4)
(c) (4, 9π/4)
(e) (−4, −3π/2)
(f) (0, 3π)
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5. In each part, a point is given in rectangular coordinates. Find two pairs of polar
coordinates for the point, one pair satisfying r ≥ 0 and 0 ≤ θ < 2π, and the second
pair satisfying r ≥ 0 and −2π < θ ≤ 0.
√
(c) (0, −2)
(a) (−5, 0)
(b) (2 3, −2)
√
(d) (−8, −8)
(e) (−3, 3 3)
(f) (1, 1)
6. In each part, find polar coordinates satisfying the stated conditions for the point whose
√
rectangular coordinates are (− 3, 1).
(a) r ≥ 0 and 0 ≤ θ < 2π
(b) r ≤ 0 and 0 ≤ θ < 2π
(c) r ≥ 0 and −2π < θ ≤ 0
(d) r ≤ 0 and −π < θ ≤ π
7 − 8 Identify the curve by transforming the given polar equation to rectangular
coordinates.
7. (a) r = 2
(c) r = 3 cos θ
8. (a) r = 5 sec θ
(c) r = 4 cos θ + 4 sin θ
(b) r sin θ = 4
(d) r =
6
3 cos θ + 2 sin θ
(b) r = 2 sin θ
(d) r = sec θ tan θ
9 − 10 Express the given equations in polar coordinates.
9.
(a) x = 3
(b) x2 + y 2 = 7
(c) x2 + y 2 + 6y = 0
(d) 9xy = 4
10. (a) y = −3
(c) x2 + y 2 + 4x = 0
(b) x2 + y 2 = 5
(d) x2 (x2 + y 2 ) = y 2
11 − 12 Use the method of Example 2.13 to sketch the curve in polar coordinates.
11. r = 2(1 + sin θ)
12. r = 1 − cos θ
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13 − 42 Sketch the curve in polar coordinates.
π
3
16. r = 4 cos θ
19. 2r = cos θ
20. r − 2 = 2 cos θ
21. r = 3(1 + sin θ) 22. r = 5 − 5 sin θ
23. r = 4 − 4 cos θ
24. r = 1 + 2 sin θ
25. r = −1 − cos θ
26. r = 4 + 3 cos θ
27. r = 2 + cos θ
28. r = 3 − sin θ
29. r = 3 + 4 cos θ
30. r − 5 = 3 sin θ
31. r = 5 − 2 cos θ
32. r = −3 − 4 sin θ
33. r 2 = cos 2θ
34. r 2 = 9 sin 2θ
35. r 2 = 16 sin 2θ
36. r = 4θ (θ ≥ 0)
37. r = 4θ (θ ≤ 0)
38. r = 4θ
39. r = −2 cos 2θ
40. r = 3 sin 2θ
41. r = 9 sin 4θ
42. r = 2 cos 3θ
17. r = 6 sin θ
14. θ = −
3π
4
15. r = 3
13. θ =
18. r = 1 + sin θ
43. Find the highest point on the cardioid r = 1 + cos θ.
44. Find the leftmost point on the upper half of the cardioid r = 1 + cos θ.
Exercise 2.3
1 − 6 Find the slope of the tangent line to the polar curve for the given value of θ.
1. r = 2 sin θ; θ = π/6
2. r = 1 + cos θ; θ = π/2
3. r = 1/θ; θ = 2
4. r = a sec 2θ; θ = π/6
5. r = sin 3θ; θ = π/4
6. r = 4 − 3 sin θ; θ = π
6 − 7 Find polar coordinates of all points at which the polar curve has a
horizontal or a vertical tangent line.
6. r = a(1 + cos θ)
7. r = a sin θ
8 − 13 Use Formula (2.8) to calculate the arc length of the polar curve.
8. The entire circle r = a
9. The entire circle r = 2a cos θ
10. The entire cardioid r = a(1 − cos θ)
11. r = sin2 (θ/2) from θ = 0 to θ = π
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12. r = e3θ from θ = 0 to θ = 2
13. r = sin3 (θ/3) from θ = 0 to θ = π/2
14. In each part, find the area of the circle by integration.
(a) r = 2a sin θ
(b) r = 2a cos θ
15 − 22 Find the area of the region described.
15. The region that is enclosed by the cardioid r = 2 + 2 sin θ.
16. The region in the first quadrant within the cardioid r = 1 + cos θ.
17. The region enclosed by the rose r = 4 cos 3θ.
18. The region enclosed by the rose r = 2 sin 2θ.
19. The region inside the circle r = 3 sin θ and outside the cardioid r = 1 + sin θ.
20. The region outside the cardioid r = 2 − 2 cos θ and inside the circle r = 4.
21. The region inside the cardioid r = 2 + 2 cos θ and outside the circle r = 3.
√
22. The region inside the rose r = 2a cos 2θ and outside the circle r = a 2.
Exercise 3.1
1. A cube of side 4 has its geometric center at the origin and its faces parallel to the
coordinate planes. Sketch the cube and give the coordinates of the corners.
2. Suppose that a box has its faces parallel to the coordinate planes and the points
(4, 2, −2) and (−6, 1, 1) are endpoints of a diagonal. Sketch the box and give the
coordinates of the remaining six corners.
3. Interpret the graph of x = 1 in the contexts of
(a) a number line
(b) 2-space
(c) 3-space
4. Find the center and radius of the sphere that has (1, −2, 4) and (3, 4, −12) as endpoints
of a diameter.
5. Show that (4, 5, 2), (1, 7, 3), and (2, 4, 5) are vertices of an equilateral triangle.
6. (a) Show that (2, 1, 6), (4, 7, 9), and (8, 5, −6) are the vertices of a right triangle.
(b) Which vertex is at the 90◦ angle?
(c) Find the area of the triangle.
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7. Find equations of two spheres that are centered at the origin and are tangent to the
sphere of radius 1 centered at (3, −2, 4).
8 − 13 Describe the surface whose equation is given.
8. x2 + y 2 + z 2 + 10x + 4y + 2z − 19 = 0
9. x2 + y 2 + z 2 − y = 0
10. 2x2 + 2y 2 + 2z 2 − 2x − 3y + 5z − 2 = 0
11. x2 + y 2 + z 2 + 2x − 2y + 2z + 3 = 0
12. x2 + y 2 + z 2 − 3x + 4y − 8z + 25 = 0
13. x2 + y 2 + z 2 − 2x − 6y − 8z + 1 = 0
14. In each part, sketch the portion of the surface that lies in the first octant.
(a) y = x
(b) y = z
(c) x = z
15. In each part, sketch the graph of the equation in 3-space.
(a) x = 1
(b) y = 1
(c) z = 1
16. In each part, sketch the graph of the equation in 3-space.
(a) x2 + y 2 = 25
(b) y 2 + z 2 = 25
(c) x2 + z 2 = 25
17. In each part, sketch the graph of the equation in 3-space.
(a) x = y 2
(b) z = x2
(c) y = z 2
18 − 27 Sketch the surface in 3-space.
18. y = sin x
19. y = ex
20. z = 1 − y 2
21. z = cos x
22. 2x + z = 3
23. 2x + 3y = 6
√
25. z = 3 − x
24. 4x2 + 9z 2 = 36
26. y 2 − 4z 2 = 4
27. yz = 1
28. If a bug walks on the sphere
x2 + y 2 + z 2 + 2x − 2y − 4z − 3 = 0
how close and how far can it get from the origin?
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29. Describe the set of all points in 3-space whose coordinates satisfy the inequality
x2 + y 2 + z 2 − 2x + 8z ≤ 8.
30. Describe the set of all points in 3-space whose coordinates satisfy the inequality
y 2 + z 2 + 6y − 4z > 3.
31. The distance between a point P (x, y, z) and the point A(1, −2, 0) is twice the distance
between P and the point B(0, 1, 1). Show that the set of all such points is a sphere,
and find the center and radius of the sphere.
Exercise 3.2
1 − 4 Sketch the vectors with their initial points at the origin.
1. (a) h2, 5i
(b) h−5, −4i
(c) h2, 0i
(e) 3i − 2j
(f) −6j
2. (a) h−3, 7i
(b) h6, −2i
(c) h0, −8i
(d) 4i + 2j
(e) −2i − j
(f) 4i
(d) −5i + 3j
3. (a) h1, −2, 2i
(c) −i + 2j + 3k
4. (a) h−1, 3, 2i
(c) 2j − k
(b) h2, 2, −1i
(d) 2i + 3j − k
(b) h3, 4, 2i
(d) i − j + 2k
−−→
5 − 6 Find the components of the vector P1 P2 .
5. (a) P1 (3, 5), P2 (2, 8)
(b) P1 (7, −2), P2 (0, 0)
(c) P1 (5, −2, 1), P2 (2, 4, 2)
6. (a) P1 (−6, −2), P2 (−4, −1)
(b) P1 (0, 0, 0), P2 (−1, 6, 1)
(c) P1 (4, 1, −3), P2 (9, 1, −3)
7. (a) Find the terminal point of v = 3i − 2j if the initial point is (1, −2).
(b) Find the terminal point of v = h−3, 1, 2i if the initial point is (5, 0, −1).
8. (a) Find the terminal point of v = h7, 6i if the initial point is (2, −1).
(b) Find the terminal point of v = i + 2j − 3k if the initial point is (−2, 1, 4).
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9 − 10 Perform the stated operations on the vectors u, v, and w.
9. u = 3i − k, v = i − j + 2k, w = 3j
(a) w − v
(b) 6u + 4w
(e) −8(v + w) + 2u
(f) 3w − (v − w)
(c) −v − 2w
(d) 4(3u + v)
10. u = h2, −1, 3i, v = h4, 0, −2i, w = h1, 1, 3i
(a) u − w
(b) 7v + 3w
(e) −3v − 8w
(f) 2v − (u + w)
(c) −w + v
(d) 3(u − 7v)
11 − 12 Find the norm of v.
11. (a) v = h1, −1i (b) v = −i + 7j
√
√
12. (a) v = h3, 4i
(b) v = 2i − 7j
(c) v = h−1, 2, 4i (d) v = −3i + 2j + k
(c) v = h0, −3, 0i (d) v = i + j + k
13. Let u = i − 3j + 2k, v = i + j, and w = 2i + 2j − 4k. Find
(a) ku + vk
(d) k3u − 5v + wk
(b) kuk + kvk
1
w
(e)
kwk
(c) k − 2uk + 2kvk
1
(f) w
kwk 14 − 15 Find the unit vectors that satisfy the stated conditions.
14. (a) Same direction as i + 4j.
(b) Oppositely directed to 6i − 4j + 2k.
(c) Same direction as the vector from the point A(−1, 0, 2) to the point B(3, 1, 1).
15. (a) Oppositely directed to 3i − 4j.
(b) Same direction as 2i − j − 2k.
(c) Same direction as the vector from the point A(−3, 2) to the point B(1, −1).
16 − 17 Find the vectors that satisfy the stated conditions.
16. (a) Oppositely directed to v = h3, −4i and half the length of v.
√
(b) Length 17 and same direction as v = h7, 0, −6i.
17. (a) Same direction as v = −2i + 3j and three times the length of v.
(b) Length 2 and oppositely directed to v = −3i + 4j + k.
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18. In each part, find the component form of the vector v in 2-space that has the stated
length and makes the stated angle θ with the positive x-axis.
(a) kvk = 3; θ = π/4
(b) kvk = 2; θ = 90◦
(c) kvk = 5; θ = 120◦
(d) kvk = 1; θ = π
19. Find the component form of v + w and v − w in 2-space, given that kvk = 1,
kwk = 1, v makes an angle of π/6 with the positive x-axis, and w makes an angle
of 3π/4 with the positive x-axis.
20. Let u = h1, 3i, v = h2, 1i, and w = h4, −1i. Find the vector x that satisfies
2u − v + x = 7x + w.
21. Let u = h−1, 1i, v = h0, 1i, and w = h3, 4i. Find the vector x that satisfies
u − 2x = x − w + 3v.
22. Find u and v if u + 2v = 3i − k and 3u − v = i + j + k.
23. Find u and v if u + v = h2, −3i and 3u + 2v = h−1, 2i.
24. In each part, find two unit vectors in 2-space that satisfy the stated condition.
(a) Parallel to the line y = 3x + 2
(b) Parallel to the line x + y = 4
(c) Perpendicular to the line y = −5x + 1
Exercise 3.3
1. In each part, find the dot product of the vectors and the cosine of the angle between
them.
(a) u = i + 2j, v = 6i − 8j
(b) u = h7, −3i, v = h0, 1i
(c) u = i − 3j + 7k, v = 8i − 2j − 2k
(d) u = h−3, 1, 2i, v = h4, 2, −5i
2. In each part use the given information to find u · v.
(a) kuk = 1, kvk = 2, the angle between u and v is π/6.
(b) kuk = 2, kvk = 3, the angle between u and v is 135◦ .
3. In each part, determine whether u and v make an acute angle, an obtuse angle, or
are orthogonal.
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(a) u = 7i + 3j + 5k, v = −8i + 4j + 2k
(b) u = 6i + j + 3k, v = 4i − 6k
(c) u = h1, 1, 1i, v = h−1, 0, 0i
(d) u = h4, 1, 6i, v = h−3, 0, 2i
4. Does the triangle in 3-space with vertices (−1, 2, 3), (2, −2, 0), and (3, 1, −4) have
an obtuse angle? Justify your answer.
5. The accompanying figure shows eight vectors that are equally spaced around a circle
of radius 1. Find the dot product of v0 with each of the other seven vectors.
v2
v3
v1
v4
v0
v5
v7
v6
6. The accompanying figure shows six vectors that are equally spaced around a circle
of radius 5. Find the dot product of v0 with each of the other five vectors.
v2
v1
v3
v0
v4
v5
7. (a) Use vectors to show that A(2, −1, 1), B(3, 2, −1), and C(7, 0, −2) are vertices of
the right triangle. At which vertex is the right angle?
(b) Use vectors to find the interior angles of the triangle with vertices (−1, 0), (2, −1),
and (1, 4).
8. (a) Show that if v = ai + bj is a vector in 2-space, then the vectors
v1 = bi + aj
and
v2 = bi − aj
are both orthogonal to v.
(b) Use the result in part (a) to find two unit vectors that are orthogonal to the vector
v = 3i − 2j. Sketch the vectors v, v1 , and v2 .
9. Explain why each of the following expressions makes no sense.
(a) u · (v · w)
(b) (u · v) + w
(c) ku · vk
(d) k · (u + v)
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10. True or false? If u · v = u · w and if u 6= 0, then v = w. Justify your conclusion.
11. Verify part (b) and (c) of Theorem 12.7 for the vectors u = 6i−j+2k, v = 2i+7j+4k,
w = i + j − 3k and k = −5.
12. Let u = h1, 2i, v = h4, −2i, and w = h6, 0i. Find
(a) u · (7v + w)
(b) k(u · w)wk
(c) kuk(v · w)
(d) (kukv) · w
13. Find r so that the vector from the point A(1, −1, 3) to the point B(3, 0, 5) is orthogonal to the vector from A to the point P (r, r, r).
14. Find two unit vectors in 2-space that make an angle of 45◦ with 4i + 3j.
15 − 16 Find the direction cosines of v.
15. (a) v = i + j − k
16.
(a) v = 3i − 2j − 6k
(b) v = 2i − 2j + k
(b) v = 3i − 4k
17. In each part, find the vector component of v along b and the vector component of v
orthogonal to b.
(a) v = 2i − j, b = 3i + 4j
(b) v = h4, 5i, b = h1, −2i
(c) v = −3i − 2j, v = 2i + j
18. In each part, find the vector component of v along b and the vector component of v
orthogonal to b.
(a) v = 2i − j + 3k, b = i + 2j + 2k
(b) v = h4, −1, 7i, b = h2, 3, −6i
(c) v = −3i − 2j, v = 2i + j
19 − 20 Express the vector v as the sum of a vector parallel to b and a vector
orthogonal to b.
19. (a) v = 2i − 4j, b = i + j
(b) v = 3i + j − 2k, b = 2i − k
(c) v = 4i − 2j + 6k, b = −2i + j − 3k
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15
20. (a) v = h−3, 5i, b = h1, 1i
(b) v = h−2, 1, 6i, b = h0, −2, 1i
(c) v = h1, 4, 1i, b = h3, −2, 5i
21. Find the work done by a force F = −3j (pounds) applied to a point that moves on
the line from (1, 3) to (4, 7), Assume that distance is measured in feet.
22. A force F = 4i − 6j + k newtons is applied to a point that moves a distance of 15
meters in the direction of the vector i + j + k. How much work is done?
23. A boat travels 100 meters due north while the wind that applies a force of 500
newtons toward the northwest. How much work does the wind do?
24. A box is dragged along the floor by a rope that applies a force of 50 lb at an angle
of 60◦ with the floor. How much work is done moving the box 15 ft?
Exercise 3.4
1. (a) Use a determinant to find the cross product
i × (i + j + k)
(b) Check your answer in part (a) by rewriting the cross product as
i × (i + j + k) = (i × i) + (i × j) + (i × k)
and evaluate each term.
2. In each part, use the two methods in Exercise 1 to find
(a) j × (i + j + k)
(b) k × (i + j + k)
3 − 6 Find u × v and check that it is orthogonal to both u and v.
3. u = h1, 2, −3i, v = h−4, 1, 2i
4. u = 3i + 2j − k, v = −i − 3j + k
5. u = h0, 1, −2i, v = h3, 0, −4i
6. u = 4i + k, v = 2i − j
7. Let u = h2, −1, 3i, v = h0, 1, 7i, and w = h1, 4, 5i. Find
(a) u × (v × w)
(b) (u × v) × w
(c) (u × v) × (v × w)
(d) (v × w) × (u × v)
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8. Find two unit vectors that are orthogonal to both
u = −7i + 3j + k,
v = 2i + 4k
9. Find two unit vectors that are normal to the plane determined by the points A(0, −2, 1),
B(1, −1, −2), and C(−1, 1, 0).
10. Find two unit vectors that are parallel to the yz-plane and are orthogonal to the vector
3i − j + 2k.
11 − 12 Find the area of the parallelogram that has u and v as adjacent sides.
11. u = i − j + 2k, v = 3j + k
12. u = 2i + 3j, v = −i + 2j − 2k
13 − 14 Find the area of the triangle with vertices P , Q, and R.
13. P (1, 5, −2), Q(0, 0, 0), R(3, 5, 1)
14. P (2, 0, −3), Q(1, 4, 5), R(7, 2, 9)
15 − 18 Find u · (v × w).
15. u = 2i − 3j + k, v = 4i + j − 3k, w = j + 5k
16. u = h1, −2, 2i, v = h0, 3, 2i, w = h−4, 1, −3i
17. u = h2, 1, 0i, v = h1, −3, 1i, w = h4, 0, 1i
18. u = i, v = i + j, w = i + j + k
19 − 20 Use a scalar triple product to find the volume of the parallelepiped that
has u, v, and w as adjacent edges.
19. u = h2, −6, 2i, v = h0, 4, −2i, w = h2, 2, −4i
20. u = 3i + j + 2k, v = 4i + 5j + k, w = i + 2j + 4k
21. In each part, use a scalar triple product to determine whether the vectors lie in the
same plane.
(a) u = h1, −2, 1i, v = h3, 0, −2i, w = h5, −4, 0i
(b) u = 5i − 2j + k, v = 4i − j + k, w = i − j
(c) u = h4, −8, 1i, v = h2, 1, −2i, w = h3, −4, 12i
MA112 Section 750001: Prepared by Dr. Archara Pacheenburawana
22. Suppose that u · (v × w). Find
(a) u · (w × v)
(b) (v × w) · u
(c) w · (u × v)
(d) v · (u × w)
(e) (u × w) · v
(f) v · (w × w)
Exercise 3.5
1 − 2 Find parametric equations for the line through P1 and P2 and also for the
line segment joining those points.
1. (a) P1 (3, −2), P2 (5, 1)
(b) P1 (5, −2, 1), P2 (2, 4, 2)
2. (a) P1 (0, 1), P2 (−3, −4)
(b) P1 (−1, 3, 5), P2 (−1, 3, 2)
3 − 4 Find parametric equations for the line whose vector equation is given.
3. (a) hx, yi = h2, −3i + th1, −4i
(b) xi + yj + zk = k + t(i − j + k)
4. (a) xi + yj = (3i − 4j) + t(2i + j)
(b) hx, y, zi = h−1, 0, 2i + th−1, 3, 0i
5 − 6 Find a point P on the line and a vector v parallel to the line by inspection.
5. (a) xi + yj = (2i − j) + t(4i − j)
(b) hx, y, zi = h−1, 2, 4i + th5, 7, −8i
6. (a) hx, yi = h−1, 5i + th2, 3i
(b) xi + yj + zk = (i + j − 2k) + tj
7 − 8 Express the given parametric equations of a line using bracket notation and
also using i, j, k notation.
7. (a) x = −3 + t, y = 4 + 5t
(b) x = 2 − t, y = −3 + 5t, z = t
8. (a) x = t, y = −2 + t
(b) x = 1 + t, y = −7 + 3t, z = 4 − 5t
9 − 16 Find parametric equations of the line that satisfies that stated conditions.
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9. The line through (−5, 2) that is parallel to 2i − 3j.
10. The line through (0, 3) that is parallel to the line x = −5 + t, y = 1 − 2t.
11. The line that is tangent to the circle x2 + y 2 = 25 at the point (3, −4).
12. The line that is tangent to the parabola y = x2 at the point (−2, 4).
13. The line through (−1, 2, 4) that is parallel to 3i − 4j + k.
14. The line through (2, −1, 5) that is parallel to h−1, 2, 7i.
15. The line through (−2, 0, 5) that is parallel to the line x = 1 + 2t, y = 4 −t, z = 6 + 2t.
16. The line through the origin that is parallel to the line x = t, y = −1 + t, z = 2.
17. Where does the line x = 1 + 3t, y = 2 − t intersect
(a) the x-axis
(b) the y-axis
(c) the parabola y = x2 ?
18. Where does the line hx, yi = h4t, 3ti intersect the circle x2 + y 2 = 25?
19 − 20 Find the intersections of the lines with xy-plane, the xz-plane, and the
yz-plane.
19. x = −2, y = 4 + 2t, z = −3 + t
20. x = 1 − 2t, y = 3 + t, z = 4 − t
21. Where does the line x = 1 + t, y = 3 − t, z = 2t intersect the cylinder x2 + y 2 = 16?
22. Where does the line x = 2 − t, y = 3t, z = −1 + 2t intersect the plane 2y + 3z = 6?
23 − 24 Show that the line L1 and L2 intersect, and find their point of intersection.
23. L1 : x = 2 + t, y = 2 + 3t, z = 3 + t
L2 : x = 2 + t, y = 3 + 4t, z = 4 + 2t
24. L1 : x + 1 = 4t, y − 3 = t, z − 1 = 0
L2 : x + 13 = 12t, y − 1 = 6t, z − 2 = 3t
25 − 26 Show that the line L1 and L2 are skew.
MA112 Section 750001: Prepared by Dr. Archara Pacheenburawana
25. L1 : x = 1 + 7t, y = 3 + t, z = 5 − 3t
L2 : x = 4 − t, y = 6, z = 7 + 2t
26. L1 : x = 2 + 8t, y = 6 − 8t, z = 10t
L2 : x = 3 + 8t, y = 5 − 3t, z = 6 + t
27 − 28 Determine whether the line L1 and L2 are parallel.
27. L1 : x = 3 − 2t, y = 4 + t, z = 6 − t
L2 : x = 5 − 4t, y = −2 + 2t, z = 7 − 2t
28. L1 : x = 5 + 3t, y = 4 − 2t, z = −2 + 3t
L2 : x = −1 + 9t, y = 5 − 6t, z = 3 + 8t
29 − 30 Determine whether the point P1 , P2 , and P3 lie on the same line.
29. P1 (6, 9, 7), P2 (9, 2, 0), P3 (0, −5, −3)
30. P1 (1, 0, 1), P2 (3, −4, −3), P3 (4, −6, −5)
Exercise 3.6
1 − 4 Find an equation of the plane that passes through the point P and has the
vector n as normal.
1. P (2, 6, 1); n = h1, 4, 2i
2. P (−1, −1, 2); n = h−1, 7, 6i
3. P (1, 0, 0); n = h0, 0, 1i
4. P (0, 0, 0); n = h2, −3, −4i
5 − 8 Find an equation of the plane indicated in the figure
19
MA112 Section 750001: Prepared by Dr. Archara Pacheenburawana
z
20
z
1
1
5.
6.
1
y
1
1
y
1
x
x
z
z
1
1
7.
8.
1
y
1
1
x
y
1
x
9 − 10 Find an equation of the plane that passes through the given point.
9. (−2, 1, 1), (0, 2, 3), and (1, 0, −1)
10. (3, 2, 1), (2, 1, −1), and (−1, 3, 2)
11 − 12 Determine whether the planes are parallel, perpendicular, or neither.
13. (a) 2x − 8y − 6z − 2 = 0
−x + 4y + 3z − 5 = 0
(b) 3x − 2y + z = 1
4x + 5y − 2z = 4
(c) x − y + 3z − 2 = 0
2x + z = 1
14. (a) 3x − 2y + z = 4
6x − 4y + 3z = 7
(b) y = 4x − 2z + 3
x = 41 y + 12 z
(c) x + 4y + 7z = 3
5x − 3y + z = 0
13 − 14 Determine whether the line and planes are parallel, perpendicular, or
neither.
13. (a) x = 4 + 2t, y = −t, z = −1 − 4t;
3x + 2y + z − 7 = 0
(b) x = t, y = 2t, z = 3t;
MA112 Section 750001: Prepared by Dr. Archara Pacheenburawana
21
x − y + 2z = 5
(c) x = −1 + 2t, y = 4 + t, z = 1 − t;
4x + 2y − 2z = 7
14. (a) x = 3 − t, y = 2 + t, z = 1 − 3t;
2x + 2y − 5 = 0
(b) x = 1 − 2t, y = t, z = −t;
6x − 3y + 3z = 1
(c) x = t, y = 1 − t, z = 2 + t;
x+y+z =1
15 − 16 Determine whether the line and planes intersect; if so, find the
coordinates of the intersection.
15. (a) x = t, y = t, z = t;
3x − 2y + z − 57 = 0
(b) x = 2 − t, y = 3 + t, z = t;
2x + y + z = 1
16. (a) x = 3t, y = 5t, z = −t;
2x − y + z + 1 = 0
(b) x = 1 + t, y = −1 + 3t, z = 2 + 4t;
x − y + 4z = 7
17 − 18 Find the acute angle of intersection of the planes.
17. x = 0 and 2x − y + z − 4 = 0
18. x + 2y − 2z = 5 and 6x − 3y + 2z = 8
19 − 28 Find an equation of the plane that satisfies the stated conditions.
19. The plane through the origin that is parallel to the plane 4x − 2y + 7z + 12 = 0.
20. The plane that contains the line x = −2+3t, y = 4+2t, z = 3−t and is perpendicular
to the plane x − 2y + z = 5.
21. The plane through the point (−1, 4, 2) that contains the line of intersection of the
planes 4x − y + z − 2 = 0 and 2x + y − 2z − 3 = 0.
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22
22. The plane through (−1, 4, −3) that is perpendicular to the line x − 2 = t, y + 3 = 2t,
and z = −t.
23. The plane through (1, 2, −1) that is perpendicular to the line of intersection of the
planes 2x + y + z = 2 and x + 2y + z = 3.
24. The plane through the points P1 (−2, 1, 4), P2 (1, 0, 3) that is perpendicular to the
planes 4x − y + 3z = 2.
25. The plane through (−1, 2, −5) that is perpendicular to the planes 2x − y + z = 1 and
x + y − 2z = 3.
26. The plane that contains the point (2, 0, 3) and the line x = −1 + t, y = t, and
z = −4 + 2t.
27. The plane whose points are equidistant from (2, −1, 1) and (3, 1, 5).
28. The plane that contains the line x = 3t, y = 1 + t, z = 2t and is parallel to the
intersection of the planes y + z = −1 and 2x − y + z = 0.
29. Find parametric equations of the line through the point (5, 0, −2) that is parallel to
the planes x − 4y + 2z = 0 and 2x + 3y − z + 1 = 0.
30. Let L be the line x = 3t + 1, y = −5t, z = t.
(a) Show that L lies in the plane 2x + y − z = 2.
(b) Show that L is parallel to the plane x + y + 2z = 0. Is the line above, below,
or on this plane?
31 − 32 Find the distance between the point and the plane.
31. (1, −2, 3); 2x − 2y + z = 4
32. (0, 1, 5); 3x + 6y − 2z − 5 = 0
33 − 34 Find the distance between parallel planes.
33. (a) −2x + y + z = 0
6x − 3y − 3z − 5 = 0
34. (b) x + y + z = 1
x + y + z = −1
35 − 36 Find the distance between the given shew lines.
35. x = 1 + 7t, y = 3 + t, z = 5 − 3t
x = 4 − t, y = 6, z = 7 + 2t
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36. x = 3 − t, y = 4 + 4t, z = 1 + 2t
x = t, y = 3, z = 2t
37. Find an equation of the sphere with center (2, 1, −3) that is tangent to the plane
x − 3y + 2z = 4.
38. Locate the point of intersection of the plane 2x + y − z = 0 and the line through
(3, 1, 0) that is perpendicular to the plane.
Exercise 3.7
1. Identify the quadric surface as an ellipsoids, hyperboloids of one sheet, hyperboloids
of two sheet, elliptic cones, elliptic paraboloids, and hyperbolic paraboloids. State
the value of a, b, and c in each case.
x2 y 2
y2
(a) z =
+
(b) z =
− x2
4
9
25
(c) x2 + y 2 − z 2 = 16
(d) x2 + y 2 − z 2 = 0
(e) 4z = x2 + 4y 2
(f) z 2 − x2 − y 2 = 1
2. Find an equation of the trace, and state whether it is an ellipse, a parabola, or a
hyperbola
1
2
(a) 4x2 + y 2 + z 2 = 4; y = 1
(b) 4x2 + y 2 + z 2 = 4; x =
(c) 9x2 − y 2 − z 2 = 16; x = 2
(d) 9x2 − y 2 − z 2 = 16; z = 2
(e) z = 9x2 + 4y 2 ; y = 2
(f) z = 9x2 + 4y 2 ; z = 4
3 − 8 Identify and sketch the quadric surface.
3. x2 +
y2 z2
+
=1
4
9
4.
x2 y 2 z 2
+
−
=1
4
9
16
5. 4z 2 = x2 + 4y 2
6. 9z 2 − 4y 2 − 9x2 = 36
7. z = y 2 − x2
8. 4z = x2 + 2y 2
Exercise 3.8
1 − 4 Convert from rectangular to cylindrical coordinates.
√
1. (4 3, 4, −4)
3. (0, 2, 0)
2. (−5, 5, 6)
√
4. (4, −4 3, 6)
MA112 Section 750001: Prepared by Dr. Archara Pacheenburawana
5 − 8 Convert from cylindrical to rectangular coordinates.
5. (4, π/6, −2)
6. (8, 3π/4, −2)
7. (5, 0, 4)
8. (7, π, −9)
9 − 12 Convert from rectangular to spherical coordinates.
√
9. (1, 3, −2)
√
11. (0, 3 3, 3)
√
10. (1, −1, 2)
√
12. (−5 3, 5, 0)
13 − 16 Convert from spherical to rectangular coordinates.
13. (5, π/6, π/4)
14. (7, 0, π/2)
15. (1, π, 0)
16. (2, 3π/2, π/2)
17 − 20 Convert from cylindrical to spherical coordinates.
√
17. ( 3, π/6, 3)
19. (2, 3π/4, 0)
18. (1, π/4, −1)
√
20. (6, 1, −2 3)
21 − 24 Convert from spherical to cylindrical coordinates.
21. (5, π/4, 2π/3)
22. (1, 7π/6, π)
23. (3, 0, 0)
24. (4, π/6, π/2)
25 − 28 An equation is given in cylindrical coordinates. Express the equation in
rectangular coordinates.
25. r = 3
26. z = r 2
27. r = 4 sin θ
28. r 2 + z 2 = 1
29 − 32 An equation is given in spherical coordinates. Express the equation in
rectangular coordinates.
29. ρ = 3
30. φ = π/4
31. ρ = 4 cos φ
32. ρ sin φ = 2 cos θ
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MA112 Section 750001: Prepared by Dr. Archara Pacheenburawana
33 − 38 An equation of a surface is given in rectangular coordinates. Find an
equation of a surface in (a) cylindrical coordinates and (b) spherical coordinates.
33. z = 3
34. z = 3x3 + 3y 2
35. x2 + y 2 = 4
36. x2 + y 2 + z 2 = 9
37. 2x + 3y + 4z = 1
38. x2 = 16 − z 2
25