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All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)
Equations of parabolas Find an equation of the following parabolas, assuming the vertex is at the origin. 20. A parabola that opens downward with directrix y = 6Equations of parabolas Find an equation of the following parabolas, assuming the vertex is at the origin. 21. A parabola with focus at (3, 0)34E35EEquations of parabolas Find an equation of the following parabolas, assuming the vertex is at the origin. 24. A parabola symmetric about the x-axis that passes through the point (1, 4)From graphs to equations Write an equation of the following parabolas. 25.From graphs to equations Write an equation of the following parabolas. 26.Equations of ellipses Find an equation of the following ellipses, assuming the center is at the origin. Sketch a graph labeling the vertices and foci. 33. An ellipse whose major axis is on the x-axis with length 8 and whose minor axis has length 6Equations of ellipses Find an equation of the following ellipses, assuming the center is at the origin. Sketch a graph labeling the vertices and foci. 34. An ellipse with vertices (6, 0) and foci (4, 0)Equations of hyperbolas Find an equation of the following hyperbolas, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, and asymptotes. Use a graphing utility to check your work. 45. A hyperbola with vertices (4, 0) and foci (6, 0)Equations of hyperbolas Find an equation of the following hyperbolas, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, and asymptotes. Use a graphing utility to check your work. 46. A hyperbola with vertices (1, 0) that passes through (53,8)Equations of ellipses Find an equation of the following ellipses, assuming the center is at the origin. Sketch a graph labeling the vertices and foci. 35. An ellipse with vertices (5, 0), passing through the point (4,35)44EEquations of hyperbolas Find an equation of the following hyperbolas, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, and asymptotes. Use a graphing utility to check your work. 47. A hyperbola with vertices (2, 0) and asymptotes y = 3x/246E47E48EFrom graphs to equations Write an equation of the following hyperbolas. 49.From graphs to equations Write an equation of the following hyperbolas. 50.51EGolden Gate Bridge Completed in 1937, San Franciscos Golden Gate Bridge is 2.7 km long and weighs about 890,000 tons. The length of the span between the two central towers is 1280 m; the towers themselves extend 152 m above the roadway. The cables that support the deck of the bridge between the two towers hang in a parabola (see figure). Assuming the origin is midway between the towers on the deck of the bridge, find an equation that describes the cables. How long is a guy wire that hangs vertically from the cables to the roadway 500 m from the center of the bridge?Eccentricity-directrix approach Find an equation of the following curves, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, asymptotes (if they exist), and directrices. Use a graphing utility to check your work. 51. An ellipse with vertices (9, 0) and eccentricity 13Eccentricity-directrix approach Find an equation of the following curves, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, asymptotes (if they exist), and directrices. Use a graphing utility to check your work. 52. An ellipse with vertices (0, 9) and eccentricity 14Eccentricity-directrix approach Find an equation of the following curves, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, asymptotes (if they exist), and directrices. Use a graphing utility to check your work. 53. A hyperbola with vertices (1, 0) and eccentricity 3Eccentricity-directrix approach Find an equation of the following curves, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, asymptotes (if they exist), and directrices. Use a graphing utility to check your work. 54. A hyperbola with vertices (0, 4) and eccentricity 257E58E59E60E61E62ETracing hyperbolas and parabolas Graph the following equations. Then use arrows and labeled points to indicate how the curve is generated as increases from 0 to 2. 61. r=11+sinTracing hyperbolas and parabolas Graph the following equations. Then use arrows and labeled points to indicate how the curve is generated as increases from 0 to 2. 62. r=11+2cosTracing hyperbolas and parabolas Graph the following equations. Then use arrows and labeled points to indicate how the curve is generated as increases from 0 to 2. 63. r=31cosTracing hyperbolas and parabolas Graph the following equations. Then use arrows and labeled points to indicate how the curve is generated as increases from 0 to 2. 64. r=112cos67EHyperbolas with a graphing utility Use a graphing utility to graph the hyperbolas r=e1+ecos, for e = 1.1, 1.3, 1.5, 1.7, and 2 on the same set of axes. Explain how the shapes of the curves vary as e changes.Tangent lines Find an equation of the tine tangent to the following curves at the given point. 69. x2 = 6y, (6, 6)70ETangent lines Find an equation of the tine tangent to the following curves at the given point. 71. y2x264=1;(6,54)Tangent lines Find an equation of the tine tangent to the following curves at the given point. 70. r=11+sin;(23,6)Tangent lines for an ellipse Show that an equation of the line tangent to the ellipse x2/a2 + y2/b2 = 1 at the point (x0, y0) is xx0a2+yy0b2=1.74E75E76EAnother construction for a hyperbola Suppose two circles, whose centers are at least 2a units apart (see figure), are centered at F1 and F2, respectively. The radius of one circle is 2a | r and the radius of the other circle is r, where r 0. Show that as r increases, the intersection points P1 and P2 of the two circles describes one branch or a hyperbola with toci at F1 and F2.The ellipse and the parabola Let R be the region bounded by the upper half of the ellipse x2/2 + y2 = 1 and the parabola y=x2/2. a. Find the area of R. b. Which is greater, the volume of the solid generated when R is revolved about the x-axis or the volume of the solid generated when R is revolved about the y-axis?Volume of an ellipsoid Suppose that the ellipse x2/a2 + y2/b2 = 1 is revolved about the x-axis. What is the volume of the solid enclosed by the ellipsoid that is generated? Is the volume different if the same ellipse is revolved about the y-axis?Area of a sector of a hyperbola Consider the region R bounded by the right branch of the hyperbola x2/a2 y2/b2 = 1 and the vertical line through the right focus. a. What is the area of R? b. Sketch a graph that shows how the area of R varies with the eccentricity e, for e 1.Volume of a hyperbolic cap Consider the region R bounded by the right branch of the hyperbola x2/a2 y2/b2 = 1 and the vertical line through the right focus. a. What is the volume of the solid that is generated when R is revolved about the x-axis? b. What is the volume of the solid that is generated when R is revolved about the y-axis?82E83E84E85E86E87E88EShared asymptotes Suppose that two hyperbolas with eccentricities e and E have perpendicular major axes and share a set of asymptotes. Show that e2 + E2 = 1.Focal chords A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties. 90. The lines tangent to the endpoints of any focal chord of a parabola y2 = 4px intersect on the directrix and are perpendicular.Focal chords A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties. 91. Let L be the latus rectum of the parabola y2 = 4px, for p 0. Let F be the focus of the parabola, P be any point on the parabola to the left of L, and D be the (shortest) distance between P and L. Show that for all P, D + |FP| is a constant. Find the constant.Focal chords A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties. 92. The length of the latus rectum of the parabola y2 = 4px or x2 = 4py is 4|p|.93E94EConfocal ellipse and hyperbola Show that an ellipse and a hyperbola that have the same two foci intersect at right angles.Approach to asymptotes Show that the vertical distance between a hyperbola x2/a2 y2/b2 = 1 and its asymptote y = bx/a approaches zero as x , where 0 b a.97E98EExplain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. A set of parametric equations for a given curve is always unique. b. The equations x = et, y = 2et, for t , describe a line passing through the origin with slope 2. c. The polar coordinates (3, 3/4) and (3, /4) describe the same point in the plane. d. The area of the region between the inner and outer loops of the limaon r = f() = 1 4 cos is 1202f()2d. e. The hyperbola y2/2 x2/4 = 1 has no x-intercept. f. The equation x2 + 4y2 2x = 3 describes an ellipse.2RE3REEliminating the parameter Eliminate the parameter to find a description of the following curves in terms of x and y. Give a geometric description and the positive orientation of the curve. 9. x = sin t = 3, y = cos t + 6; 0 t5RE6REParametric curves and tangent lines a. Eliminate the parameter to obtain an equation in x and y. b. Find the slope of the curve at the given value of t. c. Make a sketch of the curve showing the positive orientation of the curve and the tangent line at the point corresponding to the given value of t. 7. x = 8 cos t + 1, y = 8 sin t + 2, for 0 t 2; t = /3Parametric curves and tangent lines a. Eliminate the parameter to obtain an equation in x and y. b. Find the slope of the curve at the given value of t. c. Make a sketch of the curve showing the positive orientation of the curve and the tangent line at the point corresponding to the given value of t. 8. x = 4 sin 2t, y = 3 cos 2t, for 0 t ; t = /69REParametric curves a. Eliminate the parameter to obtain an equation in x and y. b. Describe the curve, indicating the positive orientation. c. Find the slope of the curve at the specified point. 10. x = t2 + 4, y = t, for 2 t 0; (5.1)Parametric curves a. Eliminate the parameter to obtain an equation in x and y. b. Describe the curve, indicating the positive orientation. c. Find the slope of the curve at the specified point. 11. x = 3 cos (t), y = 3 sin (t) 1, for 0 t ; (0, 4)12RETangent lines Find an equation of the line tangent to the cycloid x = t sin t, y = 1 cos t at the points corresponding to t = /6 and t = 2/3.Parametric descriptions Write parametric equations for the following curves. Solutions are not unique. 14. The segment of the curve x y3 + y + 1 that starts at (1, 0) and ends at (11, 2).Parametric description Write parametric equations for the following curves. Solutions are not unique. 15. The line segment from P(1, 0) to Q(1, 1) and the line segment from Q to PParametric description Write parametric equations for the following curves. Solutions are not unique. 16. The segment of the curve f(x) = x3 + 2x from (0, 0) to (2, 12)Parametric description Write parametric equations for the following curves. Solutions are not unique. 11. The circle x2 + y2 = 9, generated clockwiseParametric description Write parametric equations for the following curves. Solutions are not unique. 12. The upper half of the ellipse x29+y24=1, generated counterclockwiseArea bounded by parametric curves Find the area of the following regions. (Hint: See Exercises 103105 in section 12.1) 19. The region bounded by the y-axis and the parametric curve x = t2, y = t t3, for 0 t 1Area bounded by parametric curves Find the area of the following regions. (Hint: See Exercises 103105 in Section 12.1) 20. The region bounded by the x-axis and the parametric curve x = cos t, y = sin 2t, for 0 t /221REArc length Find the length of the following curves. 22. x = e2t sin 3t, y = e2t cos 3t; 0 t /3Arc length Find the length of the following curves. 23. x = cos 2t, y = 2t sin 2t; 0 t /424RESets in polar coordinates Sketch the following sets of points. 25. 0 r 4 and 2326RE27RE28RE29RE30REPolar curves Graph the following equations. 31. r = 3 sin 432RE33RE34REPolar conversion Write the equation r2+r(2sin6cos)=0 in Cartesian coordinates and identify the corresponding curve.Polar conversion Consider the equation r = 4/(sin + cos ). a. Convert the equation to Cartesian coordinates and identify the curve it describes. b. Graph the curve and indicate the points that correspond to = 0, /2, and 2. c. Give an interval in on which the entire curve is generated.37RE38RE39RESlopes of tangent lines a. Find all points where the following curves have vertical and horizontal tangent lines. b. Find the slope of the lines tangent to the curve at the origin (when relevant). c. Sketch the curve and all the tangent lines identified in parts (a) and (b). 30. r = 3 6 cosSlopes of tangent lines a. Find all points where the following curves have vertical and horizontal tangent lines. b. Find the slope of the lines tangent to the curve at the origin (when relevant). c. Sketch the curve and all the tangent lines identified in parts (a) and (b). 41. r = 1 sin42RE43REThe region enclosed by all the leaves of the rose r = 3 sin 445REThe region inside the limaon r = 2 + cos and outside the circle r = 2Areas of regions Find the ares of the following regions. 47. The regions inside the lemniscate r2 = 4 cos 2 and outside the circle r=248REThe area that is inside the cardioid r = 1 + cos and outside the cardioid r = 1 cos50RE51REArc length of the polar curves Find the approximate length of the following curves. 52.The limaon r = 3 + 2 cos53RE54REConic sections a. Determine whether the following equations describe a parabola, an ellipse, or a hyperbola. b. Use analytical methods to determine the location of the foci, vertices, and directrices. c. Find the eccentricity of the curve. d. Make an accurate graph of the curve. 41. y2 4x2 = 1656RE57RETangent lines Find an equation of the line tangent to the following curves at the given point. Check your work with a graphing utility. 48. x216y29=1;(203,4)Tangent lines Find an equation of the line tangent to the following curves at the given point. Check your work with a graphing utility. 45. y2=12x;(43,4)60RE61RE62RE63RE64RE65RE66REEccentricity-directrix approach Find an equation of the following curves, assuming the center is at the origin. Graph the curve, labeling vertices, foci, asymptotes (if they exist), and directrices. 55. An ellipse with vertices (0, 4) and directrices y = 1068RE69RE70RE71RE72RE73RE74RELam curves The Lam curve described by |xa|n+|yb|n=1, where a, b, and n are positive real numbers, is a generalization of an ellipse. a. Express this equation in parametric form (four pairs of equations are needed). b. Graph the curve for a = 4 and b = 2, for various values of n. c. Describe how the curves change as n increases.76REDescribe the length and direction of the vector 5v relative to v.2QC3QCGiven the points P(2.3) and Q(4, 1), find the components of PQ.Find vectors of length 10 parallel to the unit vector u35,45.Verify that the vector 513,1213 has length 1.Solve 3u | 4v = 12w for u.Interpret the following statement: Points have a location, but no size or direction; nonzero vectors have a size and direction, but no location.What is a position vector?Given a position vector v, why are there infinitely many vectors equal to v?Use the points P(3.1) and Q(7.1) to find position vectors equal to PQ and QP.If u = u1, u2 and v = v1, v2, how do you find u + v?Find two unit vectors parallel to 2,3.Is 1,1 a unit vector? Explain.Evaluate 3,1+2,4 and illustrate the sum geometrically using the Parallelogram Rule.9EExpress the vector v = v1, v2 in terms of the unit vectors i and j.How do you compute |PQ| from the coordinates of the points P and Q?The velocity of a kayak on a lake is v=2,2,22. Find the speed and heading of the kayak. Assume the positive x-axis points east and the positive y-axis points north. Assume the coordinates of v are in feet per second.Vector operations Refer to the figure and carry out the following vector operations. 17. Scalar multiples Which of the following vectors equals CE? (There may be more than one correct answer.) a. v b. 12HI c. 13OA d. u e. 12IHVector operations Refer to the figure and carry out the following vector operations. 14. Scalar multiples Which of the following vectors equal BK? (There may be more than one correct answer.) a. 6v b. 6v c. 3HI d. 3IH e. 3AOVector operations Refer to the figure and carry out the following vector operations. 19. Scalar multiples Write the following vectors as scalar multiples of u or v. a. OA b. OD c. OH d. AG e. CEVector operations Refer to the figure and carry out the following vector operations. 20. Scalar multiples Write the following vectors as scalar multiples of u or v. a. IH b. HI c. JK d. FD e. EA17EVector operations Refer to the figure and carry out the following vector operations. 22. Vector addition Write the following vectors as sums of scalar multiples of u and v. a. BF b. DE c. AF d. AD e. CD f. JD g. JI h. DB i. ILComponents and magnitudes Define the points O(0, 0), P(3, 2), Q(4, 2), and R(6, 1). For each vector, do the following. (i) Sketch the vector in an xy coordinate system. (ii) Compute the magnitude of the vector. a. OP b. QP c. RQ20EComponents and equality Define the points P(3, 1), Q(1, 2), R(1, 2), S(3, 5), T(4, 2), and U(6, 4). 25. Sketch QU, PT, and RSand the corresponding position vectors.Components and equality Define the points P(3, 1), Q(1, 2), R(1, 2), S(3, 5), T(4, 2), and U(6, 4). 26. Find the equal vectors among PQ, RS, and TU.Components and equality Define the points P(3, 1), Q(1, 2), R(1, 2), S(3, 5), T(4, 2), and U(6, 4). 27. Which of the vectors QTor SUis equal to 5, 0?Vector operations Let u = 4, 2, v = 4, 6, and w = 0, 8. Express the following vectors in the form a, b. 28. u + vVector operations Let u = 4, 2, v = 4, 6, and w = 0, 8. Express the following vectors in the form a, b. 29. w uVector operations Let u = 4, 2, v = 4, 6, and w = 0, 8. Express the following vectors in the form a, b. 30. 2u + 3vVector operations Let u = 4, 2, v = 4, 6, and w = 0, 8. Express the following vectors in the form a, b. 32. 10u 3v + wVector operations Let u = 3, 4, v = 1, 1, and w = 1, 0. 36. Find |u + v + w|.Vector operations Let u = 3, 4, v = 1, 1, and w = 1, 0. 35. Find |2v|.30EVector operations Let u = 3, 4, v = 1, 1, and w = 1, 0. 41. Which has the greater magnitude, u v or w n?Find a unit vector in the direction of v = 6,8.33E34EFind the vector v of length 6 that has the same direction as the unit vector 1/2,3/2Find the vector v that has a magnitude of 10 and a direction opposite that of the unit vector 3/5,4/5Designer vectors Find the following vectors. 73. The vector in the direction of 5, 12 with length 338EHow do you find a vector of length 10 in the direction of v = 3, 2?Let v = 8,15. a. Find a vector in the direction of v that is three times as long as v. b. Find a vector in the direction of v that has length 3.41E42EUnit vectors Define the points P(4, 1), Q(3, 4), and R(2, 6). 45. Find two unit vectors parallel to PR.44E45E46EUnit vectors a. Find two unit vectors parallel to v = 6i 8j. b. Find b if v=13,b is a unit vector. c. Find ail values of a such that w=aia3j is a unit vector.Vectors from polar coordinates Suppose O is the origin and P has polar coordinates (r, ). Show that OP = rcos,rsin.Vectors from polar coordinates Find the position vector op if o is the origin and P has polar coordinates (8, 5/6)50EFind the velocity v of an ocean freighter that is traveling 30 south of east at 30 km/hr.52EAirplanes and crosswinds Assume each plane flies horizontally in a crosswind that blows horizontally. 53. An airplane flies east to west at 320 mi/hr relative to the air in a crosswind that blows at 40 mi/hr toward the southwest (45 south of west). a. Find the velocity of the plane relative to the air va, the velocity of the crosswind w, and the velocity of the plane relative to the ground vg. b. Find the ground speed and heading of the plane relative to the ground.54EAirplanes and crosswinds Assume each plane flies horizontally in a crosswind that blows horizontally. 55. Determine the necessary air speed and heading that a pilot must maintain in order to fly her commercial jet north at a speed of 480 mi/hr relative to the ground in a crosswind that is blowing 60 south of east at 20 mi/hr.A boat in a current The water in a river moves south at 10 mi/hr. A motorboat travels due east at a speed of 20 mi/hr relative to the shore. Determine the speed and direction of the boat relative to the moving water.Another boat in a current The water in a river moves south at 5 km/hr. A motorboat travels due east at a speed of 40 km/hr relative to the water. Determine the speed of the boat relative to the shore.58EBoat in a wind A sailboat floats in a current that flows due east at 1 m/s. Due to a wind, the boats actual speed relative to the shore is 3m/s in a direction 30 north of east. Find the speed and direction of the wind.60E61E62E63E64EExplain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. Jos travels from point A to point B in the plane by following vector u, then vector v, and then vector w. If he starts at A and follows w, then v, and then u, he still arrives at B. b. Maria travels from A to B in the plane by following the vector u. By following u, she returns from B to A. c. |u + v| |u|, for all vectors u and v. d. |u + v| |u| + |v|, for all vectors u and v. e. Parallel vectors have the same length. f. If AB=CD, then A = C and B = D. g. If u and v are perpendicular, then |u + v| = |u| + |v|. h. If u and v are parallel and have the same direction, then |u + v| = |u| + |v|.Equal vectors For the points A(3, 4), B(6, 10), C(a + 2, b + 5), and D(b + 4, a 2), find the values of a and b such that AB=CD.Vector equations Use the properties of vectors to solve the following equations for the unknown vector x = a, b. Let u = 2, 3 and v = 4, 1. 63. 10x = uVector equations Use the properties of vectors to solve the following equations for the unknown vector x = a, b. Let u = 2, 3 and v = 4, 1. 64. 2x + u = v69ESolving vector equations Solve the following pairs of equations for the vectors u and v. Assume i = 1, 0 and j = 0, 1. 71. 2u + 3v = i, u v = j71E72E73EAnt on a page An ant walks due east at a constant speed of 2 mi/hr on a sheet of paper that rests on a table. Suddenly, the sheet of paper starts moving southeast at 2mi/hr. Describe the motion of the ant relative to the table.Clock vectors Consider the 12 vectors that have their tails at the center of a (circular) clock and their heads at the numbers on the edge of the clock. a. What is the sum of these 12 vectors? b. If the 12:00 vector is removed, what is the sum of the remaining 11 vectors? c. By removing one or more of these 12 clock vectors, explain how to make the sum of the remaining vectors as large as possible in magnitude. d. Consider the 11 vectors that originate at the number 12 at the top of the clock and point to the other 11 numbers. What is the sum of the vectors? (Source: Calculus, Gilbert Strang, Wellesley-Cambridge Press, 1991)Three-way tug-of-war Three people located at A, B, and C pull on ropes tied to a ring. Find the magnitude and direction of the force with which the person at C must pull so that no one moves (the system is in equilibrium).Additional Exercises 8185. Vector properties Prove the following vector properties using components. Then make a sketch to illustrate the property geometrically. Suppose u, v, and w are vectors in the xy-plane and a and c are scalars. 81. u + v = v + u Communitive propertyAdditional Exercises 8185. Vector properties Prove the following vector properties using components. Then make a sketch to illustrate the property geometrically. Suppose u, v, and w are vectors in the xy-plane and a and c are scalars. 82. (u + v) + w = u + (v + w) Associative propertyVector properties Prove the following vector properties using components. Then make a sketch to illustrate the property geometrically. Suppose u, v, and w are vectors in the xy-plane and a and c are scalars. 83. a(cv) = (ac)v Associative propertyVector properties Prove the following vector properties using components. Then make a sketch to illustrate the property geometrically. Suppose u, v, and w are vectors in the xy-plane and a and c are scalars. 84. a(u + v) = au + av Distributive property 1Vector properties Prove the following vector properties using components. Then make a sketch to illustrate the property geometrically. Suppose u, v, and w are vectors in the xy-plane and a and c are scalars. 85. (a + c)v = av + cv Distributive property 282EMagnitude of scalar multiple Prove that |cv| = |c| |v|, where c is a scalar and v is a vector.Equality of vectors Assume PQ equals RS. Does it follow that PR is equal to QS? Explain your answer.Linear independence A pair of nonzero vectors in the plane is linearly dependent if one vector is a scalar multiple of the other. Otherwise, the pair is linearly independent. a. Which pairs of the following vectors are linearly dependent and which are linearly independent: u = 2, 3, v = 12, 18, and w = 4, 6? b. Geometrically, what does it mean for a pair of nonzero vectors in the plane to be linearly dependent? Linearly independent? c. Prove that if a pair of vectors u and v is linearly independent, then given any vector w, there are constants c1 and c2 such that w = c1u + c2v.Perpendicular vectors Show that two nonzero vectors u = u1, u2 and v = v1, v2 are perpendicular to each other if u1v1 + u2v2 = 0.Parallel and perpendicular vectors Let u = a, 5 and v = 2, 6. a. Find the value of a such that u is parallel to v. b. Find the value of a such that u is perpendicular to v.The Triangle Inequality Suppose u and v are vectors in the plane. a. Use the Triangle Rule for adding vectors to explain why |u + v| |u| + |v|. This result is known as the Triangle Inequality. b. Under what conditions is |u + v| = |u| + |v|?Suppose the positive x-, y-, and z-axes point east, north, and upward, respectively. Describe the location of the points ( 1, 1, 0), (1, 0, 1), and ( 1, 1, 1) relative to the origin.To which coordinate planes are the planes x = 2 and z = 16 parallel?Describe the solution set of the equation (x 1)2 + y2 + (z + 1)2 + 4 = 0.Which of the following vectors are parallel to each other? a. u = 2, 4, 6 b. v = 4, 8, 12 c. w = 1, 2, 3Which vector has the smaller magnitude: u = 3i j k or v = 2(i + j + k)?Explain how to plot the point (3, 2, 1) in 3.What is the y-coordinate of all points in the xz-plane?Describe the plane x = 4.4ELet u = 3, 5, 7 and v = 6, 5, 1. Evaluate u + v and 3u v.What is the magnitude of a vector joining two points P(x1, y1, z1) and Q(x2, y2, z2)?Which point is farther from the origin, (3, 1, 2) or (0, 0, 4)?Express the vector from P(1, 4, 6) to Q(1, 3, 6) as a position vector in terms of i, j, and k.Points in 3 Find the coordinates of the vertices A, B, and C of the following rectangular boxes. 9.Points in 3 Find the coordinates of the vertices A, B, and C of the following rectangular boxes. 10.Points in 3 Find the coordinates of the vertices A, B, and C of the following rectangular boxes. 11.Points in 3 Find the coordinates of the vertices A, B, and C of the following rectangular boxes. 12. Assume all the edges have the same length.Plotting points in 3 For each point P(x, y, z) given below, let A(x, y, 0), B(x, 0, z), and C(0, y, z) be points in the xy-, xz-, and yz-planes, respectively. Plot and label the points A, B, C, and P in 3. 13. a. P(2, 2, 4) b. P(1, 2, 5) c. P(2, 0, 5)Plotting points in 3 For each point P(x, y, z) given below, let A(x, y, 0), B(x, 0, z), and C(0, y, z) be points in the xy-, xz-, and yz-planes, respectively. Plot and label the points A, B, C, and P in 3. 14. a. P(3, 2, 4) b. P(4, 2, 3) c. P(2, 4, 3)Sketching planes Sketch the following planes in the window [0, 5] [0, 5] [0, 5]. 15. x = 2Sketching planes Sketch the following planes in the window [0, 5] [0, 5] [0, 5]. 16. z = 3Sketching planes Sketch the following planes in the window [0, 5] [0, 5] [0, 5]. 17. y = 2Sketching planes Sketch the following planes in the window [0, 5] [0, 5] [0, 5]. 18. z = ySketching planes Sketch the following planes in the window [0, 5] [0, 5] [0, 5]. 19. The plane that passes through (2, 0, 0), (0, 3, 0), and (0, 0, 4)Sketching planes Sketch the following planes in the window [0, 5] [0, 5] [0, 5]. 20. The plane parallel to the xz-plane containing the point (1, 2, 3)Planes Sketch the plane parallel to the xy-plane through (2, 4, 2) and find its equation.22ESpheres and balls Find an equation or inequality that describes the following objects. 23. A sphere with center (1, 2, 3) and radius 4Spheres and balls Find an equation or inequality that describes the following objects. 24. A sphere with center (1, 2, 0) passing through the point (3, 4, 5)Spheres and balls Find an equation or inequality that describes the following objects. 25. A ball with center (2, 0, 4) and radius 1Spheres and balls Find an equation or inequality that describes the following objects. 26. A ball with center (0, 2, 6) with the point (1, 4, 8) on its boundaryMidpoints and spheres Find an equation of the sphere passing through P(1, 0, 5) and Q(2, 3, 9) with its center at the midpoint of PQ.Midpoints and spheres Find an equation of the sphere passing through P(4, 2, 3) and Q(0, 2, 7) with its center at the midpoint of PQ.Identifying sets Give a geometric description of the following sets of points. 29. (x 1)2 + y2 + z2 9 = 0Identifying sets Give a geometric description of the following sets of points. 30. (x + 1)2 + y2 + z2 2y 24 = 0Identifying sets Give a geometric description of the following sets of points. 31. x2 + y2 + z2 2y 4z 4 = 0Identifying sets Give a geometric description of the following sets of points. 32. x2 + y2 + z2 6x + 6y 8z 2 = 0Identifying sets Give a geometric description of the following sets of points. 33. x2 + y2 14y + z2 1334EIdentifying sets Give a geometric description of the following sets of points. 35. x2 + y2 + z2 8x 14y 18z 79Identifying sets Give a geometric description of the following sets of points. 36. x2 + y2 + z2 8x + 14y 18z 65Identifying sets Give a geometric description of the following sets of points. 37. x2 2x + y2 + 6y + z2 + 10 = 0Identifying sets Give a geometric description of the following sets of points. 38. x2 4x + y2 + 6y + z2 + 14 = 039E40E41E42E43E44EUnit vectors and magnitude Consider the following points P and Q. a. Find PQ and state your answer in two forms: a, b, c and ai + bj +ck. b. Find the magnitude of PQ. c. Find two unit vectors parallel to PQ. 45. P(1, 5, 0), Q(3, 11, 2)Unit vectors and magnitude Consider the following points P and Q. a. Find PQ and state your answer in two forms: a, b, c and ai + bj +ck. b. Find the magnitude of PQ. c. Find two unit vectors parallel to PQ. 46. P(5, 11, 12), Q(1, 14, 13)Unit vectors and magnitude Consider the following points P and Q. a. Find PQ and state your answer in two forms: a, b, c and ai + bj +ck. b. Find the magnitude of PQ. c. Find two unit vectors parallel to PQ. 47. P(3, 1, 0), Q(3, 4, 1)Unit vectors and magnitude Consider the following points P and Q. a. Find PQ and state your answer in two forms: a, b, c and ai + bj +ck. b. Find the magnitude of PQ. c. Find two unit vectors parallel to PQ. 48. P(3, 8, 12), Q(3, 9, 11)49EUnit vectors and magnitude Consider the following points P and Q. a. Find PQ and state your answer in two forms: a, b, c and ai + bj +ck. b. Find the magnitude of PQ. c. Find two unit vectors parallel to PQ. 50. P(a, b, c), Q(1, 1, 1) (a, b, and c are real numbers)Flight in crosswinds A model airplane is flying horizontally due north at 20 mi/hr when it encounters a horizontal crosswind blowing east at 20 mi/hr and a downdraft blowing vertically downward at 10 mi/hr. a. Find the position vector that represents the velocity of the plane relative to the ground. b. Find the speed of the plane relative to the ground.Another crosswind flight A model airplane is flying horizontally due east at 10 mi/hr when it encounters a horizontal crosswind blowing south at 5 mi/hr and an updraft blowing vertically upward at 5 mi/hr. a. Find the position vector that represents the velocity of the plane relative to the ground. b. Find the speed of the plane relative to the ground.Crosswinds A small plane is flying horizontally due east in calm air at 250 mi/hr when it is hit by a horizontal crosswind blowing southwest at 50 mi/hr and a 30-mi/hr updraft. Find the resulting speed of the plane and describe with a sketch the approximate direction of the velocity relative to the ground.54E55EMaintaining equilibrium An object is acted upon by the forces F1 = 10, 6, 3 and F2 = 0, 4, 9. Find the force F3 that must act on the object so that the sum of the forces is zero.Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. Suppose u and v both make a 45 angle with w in 3. Then u + v makes a 45 angle with w. b. Suppose u and v both make a 90 angle with w in 3. Then u + v can never make a 90 angle with w. c. i + j + k = 0. d. The intersection of the planes x = 1, y = 1, and z 1 is a point.