ExerciseChapter3mathstat.sci.tu.ac.th/~archara/Teaching/MA112-315/exercise112ch3… · 21. Let u=...

MA112: Prepared by Asst.Prof.Dr.Archara Pacheenburawana 1

Exercise Chapter 3

Exercise 3.1

1. A cube of side 4 has its geometric center at the origin and its faces parallel to thecoordinate 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 thecoordinates 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 endpointsof 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 thesphere of radius 1 centered at (3,−2, 4).

8− 13 Describe the surface whose equation is given.

8. x2 + y2 + z2 + 10x+ 4y + 2z − 19 = 0

9. x2 + y2 + z2 − y = 0

10. 2x2 + 2y2 + 2z2 − 2x− 3y + 5z − 2 = 0

11. x2 + y2 + z2 + 2x− 2y + 2z + 3 = 0

12. x2 + y2 + z2 − 3x+ 4y − 8z + 25 = 0

13. x2 + y2 + z2 − 2x− 6y − 8z + 1 = 0

14. If a bug walks on the sphere

x2 + y2 + z2 + 2x− 2y − 4z − 3 = 0

how close and how far can it get from the origin?

15. The distance between a point P (x, y, z) and the point A(1,−2, 0) is twice the distancebetween 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.


Answers to Exercise 3.1

2. (4, 2,−2), (4, 2, 1), (4, 1, 1), (4, 1,−2), (−6, 1, 1), (−6, 2, 1), (−6, 2,−2), (−6, 1,−2)

3. (a) point (b) line parallel to the y-axis (c) plane parallel to the yz-plane

4. radius√74, center (2, 1,−4) 6. (b) (2, 1, 6) (c) area 49

8. sphere, center (−5,−2,−1), radius 7 10. sphere, center

(

1

2,3

4,−5

4

)

, radius3√6

4

12. no graph 14. largest distance, 3 +√6; smallest distance, 3−

√6

Exercise 3.2

1− 4 Sketch the vectors with their initial points at the origin.

1. (a) 〈2, 5〉 (b) 〈−5,−4〉 (c) 〈2, 0〉

(d) −5i + 3j (e) 3i− 2j (f) −6j

2. (a) 〈−3, 7〉 (b) 〈6,−2〉 (c) 〈0,−8〉

(d) 4i+ 2j (e) −2i− j (f) 4i

3. (a) 〈1,−2, 2〉 (b) 〈2, 2,−1〉

(c) −i + 2j+ 3k (d) 2i+ 3j− k

4. (a) 〈−1, 3, 2〉 (b) 〈3, 4, 2〉

(c) 2j− k (d) i− j+ 2k

5− 6 Find the components of the vector−−→P1P2.

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 = 〈−3, 1, 2〉 if the initial point is (5, 0,−1).

8. (a) Find the terminal point of v = 〈7, 6〉 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 = 〈2,−1, 3〉, v = 〈4, 0,−2〉, w = 〈1, 1, 3〉

(a) u−w (b) 7v + 3w (c) −w + v (d) 3(u− 7v)

(e) −3v − 8w (f) 2v − (u+w)

11− 12 Find the norm of v.

11. (a) v = 〈1,−1〉 (b) v = −i+ 7j (c) v = 〈−1, 2, 4〉 (d) v = −3i+ 2j+ k

12. (a) v = 〈3, 4〉 (b) v =√2i−

√7j (c) v = 〈0,−3, 0〉 (d) v = i+ j+ k

13. Let u = i− 3j+ 2k, v = i+ j, and w = 2i+ 2j− 4k. Find

(a) ‖u+ v‖ (b) ‖u‖+ ‖v‖ (c) ‖ − 2u‖+ 2‖v‖

(d) ‖3u− 5v +w‖ (e)1

‖w‖ w (f)

∥

∥

∥

∥

1

‖w‖ w

∥

∥

∥

∥

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 = 〈3,−4〉 and half the length of v.

(b) Length√17 and same direction as v = 〈7, 0,−6〉.

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 statedlength and makes the stated angle θ with the positive x-axis.

(a) ‖v‖ = 3; θ = π/4 (b) ‖v‖ = 2; θ = 90◦

(c) ‖v‖ = 5; θ = 120◦ (d) ‖v‖ = 1; θ = π

19. Find the component form of v+w and v−w in 2-space, given that ‖v‖ = 1, ‖w‖ = 1,v makes an angle of π/6 with the positive x-axis, and w makes an angle of 3π/4 withthe positive x-axis.

20. Let u = 〈1, 3〉, v = 〈2, 1〉, and w = 〈4,−1〉. Find the vector x that satisfies 2u− v+x = 7x+w.

21. Let u = 〈−1, 1〉, v = 〈0, 1〉, and w = 〈3, 4〉. 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 = 〈2,−3〉 and 3u+ 2v = 〈−1, 2〉.

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


5. (a) 〈1, 3〉 (b) 〈−7, 2〉 (c) 〈−3, 6, 1〉 7. (a) (4,−4) (b) (8,−1,−3)

9. (a) −i+ 4j− 2k (b) 18i+ 12j− 6k (c) −i− 5j− 2k (d) 40i− 4j− 4k

(e) −2i− 16j− 18k (f) −i + 13j− 2k

11. (a)√2 (b) 5

√2 (c)

√21 (d)

√14

13. (a) 2√3 (b)

√14+

√2 (c) 2

√14+2

√2 (d) 2

√37 (e) (1/

√6)i+(1/

√6)j− (2

√6)k

(f) 1

14. (a) (−1/√17)i+ (4/

√17)j (b) (−3i + 2j− k)/

√14 (c) (4i+ j− k)/(3

√2)

16. (a)

⟨

−3

2, 2

⟩

(b)1√5〈7, 0,−6〉

18. (a) 〈3√2/2, 3

√3/2〉 (b) 〈0, 2〉 (c) 〈−5/2, 5

√3/2〉 (d) 〈−1, 0〉

20.

⟨

−2

3, 1

⟩

22. u =5

7i +

2

7j+

1

7k, v =

8

7i− 1

7j− 4

7k


24. (a) 〈1/√10, 3/

√10〉, 〈−1/

√10,−3/

√10〉 (b) 〈1/

√2,−1/

√2〉, 〈−1/

√2, 1/

√2〉

(c) ± 1√26

〈5, 1〉

Exercise 3.3

1. In each part, find the dot product of the vectors and the cosine of the angle betweenthem.

(a) u = i+ 2j, v = 6i− 8j

(b) u = 〈7,−3〉, v = 〈0, 1〉(c) u = i− 3j + 7k, v = 8i− 2j− 2k

(d) u = 〈−3, 1, 2〉, v = 〈4, 2,−5〉

2. In each part use the given information to find u · v.

(a) ‖u‖ = 1, ‖v‖ = 2, the angle between u and v is π/6.

(b) ‖u‖ = 2, ‖v‖ = 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, orare orthogonal.

(a) u = 7i+ 3j+ 5k, v = −8i+ 4j+ 2k

(b) u = 6i+ j+ 3k, v = 4i− 6k

(c) u = 〈1, 1, 1〉, v = 〈−1, 0, 0〉(d) u = 〈4, 1, 6〉, v = 〈−3, 0, 2〉

4. Does the triangle in 3-space with vertices (−1, 2, 3), (2,−2, 0), and (3, 1,−4) have anobtuse angle? Justify your answer.

5. The accompanying figure shows eight vectors that are equally spaced around a circleof radius 1. Find the dot product of v0 with each of the other seven vectors.

v0

v1

v2

v3

v4

v5

v6

v7

6. The accompanying figure shows six vectors that are equally spaced around a circle ofradius 5. Find the dot product of v0 with each of the other five vectors.


v0

v1v2

v3

v4 v5

7. (a) Use vectors to show that A(2,−1, 1), B(3, 2,−1), and C(7, 0,−2) are vertices ofthe 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 vectorv = 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) ‖u · v‖ (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 Theorem3.6 for the vectors u = 6i− j+2k, v = 2i+7j+4k,w = i+ j− 3k and k = −5.

12. Let u = 〈1, 2〉, v = 〈4,−2〉, and w = 〈6, 0〉. Find(a) u · (7v +w) (b) ‖(u ·w)w‖

(c) ‖u‖(v ·w) (d) (‖u‖v) ·w

13. Find r so that the vector from the point A(1,−1, 3) to the point B(3, 0, 5) is orthogonalto 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. (a) v = 3i− 2j− 6k (b) v = 3i− 4k

17. In each part, find the vector component of v along b and the vector component of vorthogonal to b.


(a) v = 2i− j, b = 3i+ 4j

(b) v = 〈4, 5〉, b = 〈1,−2〉(c) v = −3i− 2j, v = 2i+ j

18. In each part, find the vector component of v along b and the vector component of vorthogonal to b.

(a) v = 2i− j+ 3k, b = i+ 2j+ 2k

(b) v = 〈4,−1, 7〉, b = 〈2, 3,−6〉

19− 20 Express the vector v as the sum of a vector parallel to b and a vectororthogonal 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 = 〈−3, 5〉, b = 〈1, 1〉(b) v = 〈−2, 1, 6〉, b = 〈0,−2, 1〉(c) v = 〈1, 4, 1〉, b = 〈3,−2, 5〉

21. Find the work done by a force F = −3j (pounds) applied to a point that moves onthe 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 15meters 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 newtonstoward 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 of60◦ with the floor. How much work is done moving the box 15 ft?


1. (a) −10; cos θ = −1/√5 (b) −3; cos θ = −3/

√58 (c) 0; cos θ = 0

(d) −20; cos θ = −20/(3√70)

3. (a) obtuse (b) acute (c) obtuse (d) orthogonal

5.√2/2, 0,−

√2/2,−1,−

√2/2, 0,

√2/2 7. (a) vertex B (b) 82◦, 60◦, 38◦

13. r = 7/5 15. (a) α = β ≈ 55◦, γ ≈ 125◦ (b) α ≈ 48◦, β ≈ 132◦, γ ≈ 71◦

18. (a)

⟨

2

3,4

3,4

3

⟩

,

⟨

4

3,−7

3,5

3

⟩

(b)

⟨

−74

49,−111

49,222

49

⟩

,

⟨

270

49,62

49,121

49

⟩


20. (a) 〈1, 1〉+ 〈−4, 4〉 (b)

⟨

0,−8

5,4

5

⟩

+

⟨

−2,13

5,26

5

⟩

(c) v = 〈1, 4, 1〉 is orthogonal to b.

22. −5√3J 37. 375 ft·lb

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 = 〈1, 2,−3〉, v = 〈−4, 1, 2〉

4. u = 3i+ 2j− k, v = −i− 3j+ k

5. u = 〈0, 1,−2〉, v = 〈3, 0,−4〉

6. u = 4i+ k, v = 2i− j

7. Let u = 〈2,−1, 3〉, v = 〈0, 1, 7〉, and w = 〈1, 4, 5〉. 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 thevector 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 = 〈1,−2, 2〉, v = 〈0, 3, 2〉, w = 〈−4, 1,−3〉

17. u = 〈2, 1, 0〉, v = 〈1,−3, 1〉, w = 〈4, 0, 1〉

18. u = i, v = i + j, w = i+ j + k

19− 20 Use a scalar triple product to find the volume of the parallelepiped thathas u, v, and w as adjacent edges.

19. u = 〈2,−6, 2〉, v = 〈0, 4,−2〉, w = 〈2, 2,−4〉

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 thesame plane.

(a) u = 〈1,−2, 1〉, v = 〈3, 0,−2〉, w = 〈5,−4, 0〉(b) u = 5i− 2j+ k, v = 4i− j + k, w = i− j

(c) u = 〈4,−8, 1〉, v = 〈2, 1,−2〉, w = 〈3,−4, 12〉

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)


1. (a) −j+ k 3. 〈7, 10, 9〉 5. 〈−4,−6,−3〉

7. (a) 〈−20,−67,−9〉 (b) 〈−78, 52,−26〉 (c) 〈0,−56,−392〉 (d) 〈0, 56, 392〉

9. ± 1√6〈2, 1, 1〉 11.

√59 13.

√374/2 15. 80 17. −3 19. 16

21. (a) yes (b) yes (c) no


Exercise 3.5

1− 2 Find parametric equations for the line through P1 and P2 and also for theline 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) 〈x, y〉 = 〈2,−3〉+ t〈1,−4〉(b) xi + yj+ zk = k + t(i− j+ k)

4. (a) xi+ yj = (3i− 4j) + t(2i+ j)

(b) 〈x, y, z〉 = 〈−1, 0, 2〉+ t〈−1, 3, 0〉

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) 〈x, y, z〉 = 〈−1, 2, 4〉+ t〈5, 7,−8〉

6. (a) 〈x, y〉 = 〈−1, 5〉+ t〈2, 3〉(b) xi + yj+ zk = (i + j− 2k) + tj

7− 8 Express the given parametric equations of a line using bracket notation andalso 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.

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 + y2 = 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 〈−1, 2, 7〉.

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 〈x, y〉 = 〈4t, 3t〉 intersect the circle x2 + y2 = 25?

19− 20 Find the intersections of the lines with xy-plane, the xz-plane, and theyz-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 + y2 = 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

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)


1. (a) x = 3 + 2t, y = −2 + 3t; line segment: 0 ≤ t ≤ 1

(b) x = 5− 3t, y = −2 + 6t, z = 1 + t; line segment: 0 ≤ t ≤ 1

3. (a) x = 2 + t, y = −3 − 4t (b) x = t, y = −t, z = 1 + t

5. (a) P (2,−1), v = 4 i− j (b) P (−1, 2, 4), v = 5 i+ 7 j− 8k

7. (a) 〈−3, 4〉+ t〈1, 5〉; −3 i+ 4 j+ t(i + 5 j)

(b) 〈2,−3, 0〉+ t〈−1, 5, 1〉; 2 i− 3 j+ t(−i + 5 j+ k)

9. x = −2 + 2t, y = 2− 3t 11. x = 3 + 4t, y = −4 + 3t

13. x = −1 + 3t, y = 2− 4t, z = 4 + t 15. x = −2 + 2t, y = −t, z = 5 + 2t

17. (a) x = 7 (b) y =7

3(c) x =

−1 ±√85

6, y =

43∓√85

18

19. (−2, 10, 0); (−2, 0,−5); the line does not intersect the yz-plane.

21. (0, 4,−2); (4, 0, 6) 23. (1,−1, 2) 27. The lines are parallel.

29. The points do not lie on the same line.

Exercise 3.6

1− 4 Find an equation of the plane that passes through the point P and has thevector n as normal.

1. P (2, 6, 1); n = 〈1, 4, 2〉

2. P (−1,−1, 2); n = 〈−1, 7, 6〉

3. P (1, 0, 0); n = 〈0, 0, 1〉

4. P (0, 0, 0); n = 〈2,−3,−4〉

5− 8 Find an equation of the plane indicated in the figure


5.y

1

z

1

x

1

6.y

1

z

1

x

1

7.y

1

z

1

x

1

8.y

1

z

1

x

1

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.

11. (a) 2x− 8y − 6z − 2 = 0−x+ 4y + 3z − 5 = 0

(b) 3x− 2y + z = 14x+ 5y − 2z = 4

(c) x− y + 3z − 2 = 02x+ z = 1

12. (a) 3x− 2y + z = 46x− 4y + 3z = 7

(b) y = 4x− 2z + 3x = 1

4y + 1

2z

(c) x+ 4y + 7z = 35x− 3y + z = 0

13− 14 Determine whether the line and planes are parallel, perpendicular, orneither.

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 thecoordinates 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 perpendicularto the plane x− 2y + z = 5.

21. The plane through the point (−1, 4, 2) that contains the line of intersection of theplanes 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 theplanes 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 theplanes 4x− y + 3z = 2.

25. The plane through (−1, 2,−5) that is perpendicular to the planes 2x− y+ z = 1 andx+ y − 2z = 3.

26. The plane that contains the point (2, 0, 3) and the line x = −1 + t, y = t, andz = −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 theintersection 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 tothe 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, oron 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 = 06x− 3y − 3z − 5 = 0

34. (b) x+ y + z = 1x+ 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 planex− 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.



1. x+ 4y + 2z = 28 3. z = 0 5. x− y = 0 7. y + z = 1 9. 2y − z = 1

11. (a) parallel (b) perpendicular (c) neither

13. (a) parallel (b) neither (c) perpendicular

15. (a) point of intersection is(

5

2, 5

2, 52

)

(b) no intersection

17. 35◦ 19. 4x− 2y + 7z = 0 21. 4x− 13y + 21z = −14 23. x+ y − 3z = 6

25. x+ 5y + 3z = −6 27. x+ 2y + 4z = 29

229. x = 5− 2t, y = 5t, z = −2 + 11t

31.5

333. 5/

√54 35. 25/

√126 37. (x− 2)2 + (y − 1)2 + (z + 3)2 = 121

14

Exercise 3.7

1. Identify the quadric surface as an ellipsoids, hyperboloids of one sheet, hyperboloidsof two sheet, elliptic cones, elliptic paraboloids, and hyperbolic paraboloids. State thevalue of a, b, and c in each case.

(a) z =x2

4+

y2

9(b) z =

y2

25− x2

(c) x2 + y2 − z2 = 16 (d) x2 + y2 − z2 = 0

(e) 4z = x2 + 4y2 (f) z2 − x2 − y2 = 1

2. Find an equation of the trace, and state whether it is an ellipse, a parabola, or ahyperbola

(a) 4x2 + y2 + z2 = 4; y = 1 (b) 4x2 + y2 + z2 = 4; x = 1

2

(c) 9x2 − y2 − z2 = 16; x = 2 (d) 9x2 − y2 − z2 = 16; z = 2

(e) z = 9x2 + 4y2; y = 2 (f) z = 9x2 + 4y2; z = 4

3− 8 Identify and sketch the quadric surface.

3. x2 +y2

4+

z2

9= 1 4.

x2

4+

y2

9− z2

16= 1

5. 4z2 = x2 + 4y2 6. 9z2 − 4y2 − 9x2 = 36

7. z = y2 − x2 8. 4z = x2 + 2y2



1. (a) elliptic paraboloid, a = 2, b = 3

(b) hyperbolic paraboloid, a = 1, b = 5

(c) hyperboloid of one sheet, a = b = c = 4

(d) circular cone, a = b = 1

(e) elliptic paraboloid, a = 2, b = 1

(f) hyperboloid of two sheets, a = b = c = 1

3. Ellipsoid 4. Hyperboloid of one sheet 5. Elliptic cone

6. Hyperboloid of two sheets 7. Hyperbolic paraboloid 8. Elliptic paraboloid

Extra Problems

1. If a bug walks on the sphere

x2 + y2 + z2 + 2x− 2y − 4z − 3 = 0

how close and how far can it get from the origin?

Solution the closest distance:√6− 3, the farthest distance:

√6 + 3

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?

(b) What is the maximum distance between a point on A and a point on B?

Solution (a) 4 (b) 18

3. What is the equation of the sphere with center (−1, 2, 3) which passes through thepoint (0, 4, 1)?

Solution (x+ 1)2 + (y − 2)2 + (z − 3)2 = 9

4. A sphere has equation x2 + y2 + z2 = 10y − 16z + C, where C is a constant.

(a) Find the center of the sphere.

(b) Find the radius of the sphere in terms of C.

(c) If the radius of the sphere is equal to 10, find the points where the sphereintersects the y-axis.

Solution (a) (0, 5,−8) (b)√89 + C (c) (0, 11, 0) and (0,−1, 0)


5. Find the equation of the sphere with center (1, 2, 4) that touches the xz-plane.

6. (a) Given that the mid-point of the line segment from (x1, y1, z1) to (x2, y2, z2) is

(

x1 + x2

2,y1 + y2

2,z1 + z2

2

)

.

Find the equation of a sphere if one of its diameter has end-points P (3, 5,−3) andQ(7, 3, 1).

(b) If the angle between two unit vectors u and v isπ

3, then find the value of ‖2u−3v‖.

Solution (a) (x− 5)2 + (y − 4)2 + (z + 1)2 = 9 (b)√7

7. Two vectors u and v are of the form u = a(−i + j) and v = bi + j where a and b arescalars. If u+ v + 10i− j = 0, find the scalars a and b.

Solution a = 0 and b = −10

8. Given two vectors u = 〈−3, 2, 2〉 and v = 〈4, 3,−1〉.

(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.

Solution (a)

⟨

− 3√17

,2√17

,2√17

⟩

(b) θ = cos−1

(

− 8√442

)

(c) cosα = − 3√17

, cos β =2√17

, cos γ =2√17

(d)

⟨

−16

13,−12

13,4

13

⟩

9. Let a = 〈−1, 4, 1〉 and b = 〈1, 0,−3〉 be two vectors in 3-space.

(a) Find the orthogonal projection of b onto a.

(b) Find the angle between the vectors a and b.

(c) If r = 〈x, y, z〉, show that the vector equation (r − a) · (r − b) = 0 represents asphere. Then find the center and radius of this sphere.

(d) Find the area of the parallelogram that has a and b as adjacent sides.

(e) Find an equation of the plane that passes through the point (2,−1, 3) and con-tains vectors a and b.

Solution (a)

⟨

2

9,−8

9,−2

9

⟩

(b) cos−1

(

− 4√180

)

(c) sphere: radius 3 and center (0, 2,−1) (d)√164

(e) −12(x− 2)− 2(y + 1)− 4(z − 3) = 0

10. Let a =⟨√

2, 1, 1⟩

and b =⟨

−√2, 4,−1

⟩

be two vectors in 3-space.


(a) Find the orthogonal projection of b onto a.

(b) Find the angle between the vectors a and a+ b.

Solution (a) 1

4

⟨√2, 1, 1

⟩

(b) π

3

11. Given u = i− 2 j+ k and v = −i + j.

(a) Find the orthogonal projection of u onto v.

(b) Find the vector component of u orthogonal to v.

12. Find a vector a such that the norm of the projection of a on b is 2, where b =〈4,−2,−4〉.(There are several a satisfying the condition above.)

13. Find a vector that has the opposite direction to the vector v = 〈−1, 0,−3〉 with themagnitude of 3.

14. 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.

(b) Find the area of triangle PQR.

15. 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 ‖a×b‖ = ‖a‖‖b‖.Which of the following statements must be true? (Simply circle all the statementsthat must be true—no explanations are necessary.)

i. a 6= b

ii. a and b are parallel

iii. a is perpendicular to b

iv. either a or b must be a unit vector

Solution (a) 〈24, 24,−36〉(b) This is zero. a× b is a vector perpendicular to a (and b, not that it matters), soa · (a× b) = 0.

(c) This doesn’t exist because a ·b is a scalar (a number), and we can’t cross a vectorwith a scalar.

16. Let a,b, and c be vectos in R3 with (c× b) · a = −4. Find the value of

a ·[(

b+ c)

×(

c+ a)]

.


17. A force F = 4i − 6j + k newtons is applied to a point that moves a distance of 15meters in the direction of the vector i + j+ k. How much work is done?

Solution − 30√3+ 1

18. A tow truck drags a stalled car along a road. The chain makes an angle of 30◦ or π/6radians with the road. The tension (force) in the chain is 1500 N. How much work isdone by the truck in pulling the car 1 km; i.e., 1000 m?

Solution 750, 000√3 joules

19. A worker pushes a box up a ramp that is inclined at an angle of 45 degrees above thehorizontal. While doing it he exerts a force of 10 Newtons directed at an angle of 15degrees above the horizontal. Find the distance in meters traveled by the box if thework done on it is 15 Newton-meters.

Solution√3

20. 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.

Solution (a) 2700 lb (b) 5400

(√3

2

)

lb

21. 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 PQ and PR.

(a) Find the area of the parallelogram.

(b) Find the cosine of the angle between the vectors−→PQ and

−→PR.

(c) Find the orthogonal projection of−→PQ on

−→PR.

Solution (a) 7 (b)4√65

(c)

⟨

12

13,8

13, 0

⟩

22. Consider a triangle with vertices A(0, 0, a), B(1, 1, 0), and C(2, 3, 0).

(a) Compute the area of the given triangle.

(b) Find all values of a such that the area of the triangle is equal to 8.

Solution (a) 1

2

√5a2 + 1 (b) ±

√51

23. Find the vector component of v = 2i − j + 3k along b = i + 2j + 2k and the vectorcomponent of v orthogonal to b.

Solution2

3i+

4

3j+

4

3k,

4

3i− 7

3j+

5

3k


24. Consider a triangle with vertices P (1, 1, 2), Q(2, 2, 0), and R(2, 3, 5).

(a) Find the orthogonal projection of−→PQ onto

−→PR.

(b) Find the area of the given triangle.

(c) Find parametric equations of the line perpendicular to the triangle and passingthrough the point R.

Solution (a) − 3

14〈1, 2, 3〉 (b)

1

2

√75 (c) x = 2 + 7t, y = 3− 5t, z = 5 + t

25. Find the volume of the parallelepiped with three adjacent edges formed by u =〈1, 0, 2〉, v = 〈0, 2, 3〉, and w = 〈0, 1, 3〉.Solution 3

26. 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.

Solution 41

27. 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.

Solution skew line

28. Find parametric equations of the line L passing through the points P (0, 2, 1) andQ(2, 0, 2).

Solution x = 2 + 2t, y = −2t, z = 2 + t

29. Find parametric equations of the line through the point (2,−1, 3) which is perpendic-ular to the vectors a = 〈1,−2, 3〉 and b = 〈3,−2, 1〉.Solution x = 2 + 4t, y = −1 + 8t, z = 3 + 4t

30. 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.

Solution skew line

31. Find parametric equations of the line L that passed through the point (−2, 0, 5) andparallel to the line x = 1 + 2t, y = 4− t, z = 6 + 2t.

Solution x = −2 + 2t, y = −t, z = 5 + 2t


32. (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 .)

Solution (a) −10x− 5y − 10z + 65 = 0 (b) 1 (c)(

, 1

3,−1

3,−1

3

)

33. Two objects, O1 and O2, start to travel at the same time along paths defined by theparametric equations:

L1 : x = 2 + 2t, y = 5− t, z = 1− 5t,

and

L2 : x = 2s, y = −6 + s2, z = −1

3s3,

respectively, where t, s ≥ 0.

(a) Find a common point of the two objects.

(b) Do O1 and O2 collide? (Give a sufficient reason)

34. For each of the following problems, determine whether the planes P1 and P2 areparallel. Explain your answer. If the planes are parallel, find the distance betweenthem.

(a) P1 : x+ 2y − 4z = 2 ; P2 : 2x+ y − 4z = 1

(b) P1 : x+ 2y − 4z = 2 ; P2 : −2x− 4y + 8z = 1

35. Let 2x− y + 3z − 4 = 0 be the equation of the plane P and let Q(−1, 0, 4) be pointin 3-space.

(a) Find parametric equations of the line passing through the point Q and perpen-dicular to the plane P .

(b) Find an equation for the plane through the point Q and parallel to the plane P .

(c) Find the acute angle of intersection between the plane P and the plane definedby −3x− 2y − z + 6 = 0.

(d) Find the distance from the point Q to the plane P .

Solution (a) x = −1+2t, y = −t, z = 4+3t (b) 2(x+1)−1(y−0)+3(z−4) = 0

(c)π

3(d)

6√14

36. Let x = 1 − t, y = t, z = 2 + 4t be the parametric equations of the line L and letP (1, 2, 3) be point in 3-space.


(a) Find the equation for the plane containing the point P and the line L.

(b) Find the distance between the plane from part (a) and the plane 2x−y+3z = 7.

Solution (a) 7(x− 1)− (y − 2) + 2(z − 3) = 0 (b) undefined

37. Find the equation of the plane that is equidistant from the points A = (3, 2, 1) andB = (−3,−2,−1) (that is, every point on the plane has the same distance from thetwo given points).

Solution −6x− 4y − 2z = 0

38. (a) Given two planes ax + by + cz + d = 0 and ex + fy + gz + h = 0, when each ofa, b, c, d, e, f, g, h is any real number. Give conditions in order that the planes do notintersect to each other.

(b) Do the planes x + 2y − 3z + 5 = 0 and 4x − 12y − 28z − 4 = 0 intersect? If so,find parametric equations of the line of intersection.

39. For the surface x2 + y2 − 4z2 − 2y = α.

(a) If α = 0, what quadric surface is this? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(b) If α > 0, what quadric surface is this? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Solution (a) Hyperboloids of One Sheet (b) Hyperboloids of One Sheet

40. Given the surface ax2 + by2 + cz2 = 1.

(a) Classify the quadric surface if a, b, c > 0.

(b) Classify the quadric surface if only one of a, b, c is positive.

Solution (a) Ellipsoids (b) Hyperboloids of Two Sheet

41. Match each equation with its graph.

(a) z − x2 − y2 = 0

(b) z2 − x2 − y2 = 1

(c) z2 − x2 − y2 = 0

(I)

x

z

y

(II)

x

z

y


(III)

x

z

y

(IV )

x

z

y

Solution (a) (I) (b) (IV) (c) III

42. Identify and sketch the surface of the equation 36y2 − 9x2 − 4z2 = 36.

43. Match each graph with one of the equations.

A. z2 = x2 +y2

4

B. z2 − x2 − y2

4= 1

C. z = x2 +y2

4

D. z =y2

4− x2

x

z

y Solution C.

x

z

y Solution A.

ExerciseChapter3mathstat.sci.tu.ac.th/~archara/Teaching/MA112-315/exercise112ch3… · 21. Let u=...

Documents

Transcript of ExerciseChapter3mathstat.sci.tu.ac.th/~archara/Teaching/MA112-315/exercise112ch3… · 21. Let u=...