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MA112: Prepared by Dr. Archara Pacheenburawana
Exercise Chapter 3
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.
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
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MA112: Prepared by Dr. Archara Pacheenburawana
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?
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
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MA112: Prepared by Dr. Archara Pacheenburawana
−−→
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).
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
(c) −v − 2w
(d) 4(3u + v)
(e) −8(v + w) + 2u (f) 3w − (v − w)
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).
MA112: Prepared by Dr. Archara Pacheenburawana
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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.
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
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MA112: Prepared by Dr. Archara Pacheenburawana
(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.
(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).
MA112: Prepared by Dr. Archara Pacheenburawana
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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)
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
(b) v = 2i − 2j + k
16.
(b) v = 3i − 4k
(a) v = 3i − 2j − 6k
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
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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
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
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MA112: Prepared by Dr. Archara Pacheenburawana
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)
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.
MA112: Prepared by Dr. Archara Pacheenburawana
(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
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
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9 − 16 Find parametric equations of the line that satisfies that stated conditions.
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.
25. L1 : x = 1 + 7t, y = 3 + t, z = 5 − 3t
L2 : x = 4 − t, y = 6, z = 7 + 2t
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MA112: Prepared by Dr. Archara Pacheenburawana
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
z
z
1
1
5.
6.
1
y
1
1
y
1
x
x
z
z
1
1
7.
8.
1
y
1
1
x
1
x
y
MA112: Prepared by Dr. Archara Pacheenburawana
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;
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.
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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.
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.
MA112: Prepared by Dr. Archara Pacheenburawana
14
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
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.
y2
x2 y 2
+
(b) z =
− x2
(a) z =
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
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MA112: Prepared by Dr. Archara Pacheenburawana
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.
y2 z2
+
=1
3. x +
4
9
x2 y 2 z 2
4.
+
−
=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
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)
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.
MA112: Prepared by Dr. Archara Pacheenburawana
√
17. ( 3, π/6, 3)
18. (1, π/4, −1)
√
20. (6, 1, −2 3)
19. (2, 3π/4, 0)
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
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
Extra Problems
1. 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?
2. Write the equations of the following two spheres:
• Sphere A: center (2, −3, 4) and radian 3,
• Sphere B: center (4, 3, −5) and radian 4.
(a) What is the minimum distance between a point on A and a point on B?
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MA112: Prepared by Dr. Archara Pacheenburawana
17
(b) What is the maximum distance between a point on A and a point on B?
3. What is the equation of the sphere with center (−1, 2, 3) which passes through the
point (0, 4, 1)?
4. Find the equation of the sphere with center (1, 2, 4) that touches the xz-plane.
5. Two vectors u and v are of the form u = a(−i + j) and v = bi + j where a and b are
scalars. If u + v + 10i − j = 0, find the scalars a and b.
6. Find the vector component of v = 2i − j + 3k along b = i + 2j + 2k and the vector
component of v orthogonal to b.
7. Find a vector a such that the norm of the projection of a on b is 2, where b =
h4, −2, −4i.
(There are several a satisfying the condition above.)
8. 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?
9. Given two vectors u = h−3, 2, 2i and v = h4, 3, −1i.
(a) Find a unit vector in the same direction as u.
(b) Find the angle between u and v.
(c) Find the direction cosines of u.
(d) Find the orthogonal projection of u on v.
10. A tow truck drags a stalled car along a road. The chain makes an angle of 30◦ or π/6
radians with the road. The tension (force) in the chain is 1500 N. How much work is
done by the truck in pulling the car 1 km; i.e., 1000 m?
11. A sport utility vehicle with a gross weight of 5400 pounds is parked on a slope of 30◦ .
Assume that the only force the only force to overcome is the force of gravity.
(a) Find the force required to keep the vehicle from rolling down the hill.
(b) Find the force perpendicular to the hill.
12. Three of the four vertices of a parallelogram are P (0, −1, 1), Q(0, 1, 0), and R(3, 1, 1).
Two of the sides are P Q and P R.
(a) Find the area of the parallelogram.
−→
−→
(b) Find the cosine of the angle between the vectors P Q and P R.
−→
−→
(c) Find the orthogonal projection of P Q on P R.
13. Given points P (1, 0, 0), Q(0, 2, 0), and R(0, 0, 3) in 3-space.
(a) Find a vector orthogonal to the plane passing through P, Q, and R.
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MA112: Prepared by Dr. Archara Pacheenburawana
(b) Find the area of triangle P QR.
14. Use a scalar triple product to find the volume of the parallelepiped that has u =
3i + j + 2k, v = 4i + 5j + k, and w = i + 2j + 4k as adjacent edges.
15. Find the volume of the parallelepiped with three adjacent edges formed by u =
h1, 0, 2i, v = h0, 2, 3i, and w = h0, 1, 3i.
16. Let a and b be two vectors such that a × b = 2i + 2j − 3k.
(a) What is (−3b) × (4a)?
(b) Find a · (a × b) if it exists. If it doesn’t exist then explain why not.
(c) Find a × (a · b) if it exists. If it doesn’t exist then explain why not.
(d) Suppose that not only does a × b = 2i + 2j − 3k, but also that ka × bk = kakkbk.
Which of the following statements must be true?
i.
ii.
iii.
iv.
a 6= b
a and b are parallel
a is perpendicular to b
either a or b must be a unit vector
17. Find parametric equations of the line L passing through the points P (0, 2, 1) and
Q(2, 0, 2).
18. Find an equation of the plane through the points P1 (−2, 1, 1), P2 (0, 2, 3), and P3 (1, 0, −1).
19. Given a point P (0, 1, 2) and the vectors u = h1, 0, 1i and v = h2, 3, 0i, find
(a) an equation for the plane that contains P and whose normal vector is perpendicular to the two vectors u and v.
(b) Find the distance from the point (2, 1, −1) to the plane from part (a).
20. Find the distance between parallel planes
−2x + y + z = 0 and 6x − 3y − 3z − 5 = 0
21. Find parametric equations of the line L that passed through the point (−2, 0, 5) and
parallel to the line x = 1 + 2t, y = 4 − t, z = 6 + 2t.
22. Are the lines L and M given below parallel or skew or intersect? Explain your answer.
If they intersect, find the intersection point.
L : x = 1 − 2t,
y = 2t,
z =5−t
M : x = 3 + 2s,
y = −2,
z = 3 + 2s
where s and t are parameters.
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MA112: Prepared by Dr. Archara Pacheenburawana
23. Let L1 and L2 be the lines
L1 : x = 1 + 2t,
y = 3t,
z = 2−t
L2 : x = −1 + t,
y = 4 + t, z = 1 + 3t
Determine whether the lines L1 and L2 given above are parallel, skew or intersecting.
24. (a) Find an equation of the plane containing the lines
x = 4 − 4t,
y = 3 − t,
z = 1 + 5t and
x = 4 − t,
y = 3 + 2t,
z=1
(b) Find the distance from the point (1, 0, −1) to the plane 2x + y − 2z = 1.
(c) Find the point P in the plane 2x+y−2z = 1 which is closest to the point (1, 0, −1).
(Hint: You can use part (b) of this problem to help find P .)
25. Let x − z = 1 be the equation of the plane P1 , y + 2z = 3 be the equation of the
plane P2 , and x + y − 2z = 1 be the equation of the plane P3 . Let L be the line of
intersection of the planes P1 and P2 .
(a) What is the normal vector to the plane P1 ?
(b) Find the acute angle of intersection between the planes P1 and P2 .
(c) Find the parametric equations of the line L.
(d) Find the equation of the plane P passes through the line L and is perpendicular
to the plane P3 .
26. Find the spherical coordinates of the point that has rectangular coordinates
√ (x, y, z) = 1, −1, 2
√
π
as a point in spherical coordinates.
2
√
π
2, π, −
(a) The rectangular coordinates of
are . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
√
π
are . . . . . . . . . . . . . . . . . . . . . . . . . . .
(b) The cylindrical coordinates of
2, π, −
2
8. Given
2, π, −
27. (a) Find √the spherical coordinates of the point that has rectangular coordinates
−1, 1, − 2 .
(b) An equation of a surface x2 = 16 − z 2 is given in rectangular coordinates. Find
an equation of the surface in cylindrical coordinates.
28. The equation of the hyperboloid of one sheet is of the kind:
x2 + y 2 = 1 + z 2 .
Rewrite this equation using the cylindrical and spherical coordinate systems.
MA112: Prepared by Dr. Archara Pacheenburawana
29. For the surface x2 + 4y 2 + z 2 − 2x = n.
(a) If n = 0, what quadric surface is this? Explain.
(b) If n > 0, what quadric surface is this? Explain.
20