Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Precalculus
For Exercises 73-76, determine whether the two ordered pairs in polar coordinates represent the same point in the plane. If not, explain the change needed to make the two ordered pairs represent the same point. Assume that n is any integer. r, and r,74PEFor Exercises 73-76, determine whether the two ordered pairs in polar coordinates represent the same point in the plane. If not, explain the change needed to make the two ordered pairs represent the same point. Assume that n is any integer. r, and r,For Exercises 73-76, determine whether the two ordered pairs in polar coordinates represent the same point in the plane. If not, explain the change needed to make the two ordered pairs represent the same point. Assume that n is any integer. r, and r,+2nFor Exercises 77-78, determine whether the statement is true or false for two points Pr1,1 and Qr2,2 represented in polar coordinates. If the statement is false, give a counterexample. If r1=r2 then P and Q are the same distance from the pole.For Exercises 77-78, determine whether the statement is true or false for two points Pr1,1 and Qr2,2 represented in polar coordinates. If the statement is false, give a counterexample. If 1=2 then P and Q are the same distance from the pole.For Exercises 79-82, a point in polar coordinates is given. Let n be an integer and find all other polar representations of the given point. 3,480PEFor Exercises 79-82, a point in polar coordinates is given. Let n be an integer and find all other polar representations of the given point. 4,56For Exercises 79-82, a point in polar coordinates is given. Let n be an integer and find all other polar representations of the given point. 6,74For Exercises 83-88, describe the graph of the set of points r, represented by the polar inequality. Assume that the polar axis is oriented to coincide with the positive x-axis in a rectangular coordinate system. r8For Exercises 83-88, describe the graph of the set of points r, represented by the polar inequality. Assume that the polar axis is oriented to coincide with the positive x-axis in a rectangular coordinate system. r5For Exercises 83-88, describe the graph of the set of points r, represented by the polar inequality. Assume that the polar axis is oriented to coincide with the positive x-axis in a rectangular coordinate system. 2r4For Exercises 83-88, describe the graph of the set of points r, represented by the polar inequality. Assume that the polar axis is oriented to coincide with the positive x-axis in a rectangular coordinate system. 1r2For Exercises 83-88, describe the graph of the set of points r, represented by the polar inequality. Assume that the polar axis is oriented to coincide with the positive x-axis in a rectangular coordinate system. 04For Exercises 83-88, describe the graph of the set of points r, represented by the polar inequality. Assume that the polar axis is oriented to coincide with the positive x-axis in a rectangular coordinate system. 2Suppose that seating in a theater is in an area defined in polar coordinates where the pole is located at the front and center of the stage labeled as point A. The seating area is defined by 44 and 30r100 , and the values of r are in feet. a. Sketch the seating area. b. Determine the amount of area for seating. Write the exact answer in terms of and give an approximation to the nearest square foot.A rotating sprinkler spreads water over an area defined by 656 and 4r24 relative to the sprinkler head at the pole at point A and an imaginary line defined by =0 . The values of r are in feet. a. Sketch the area that is watered. b. Determine the amount of area for watered. Write the exact answer in terms of and give an approximation to the nearest square foot.A player in a video game must knock out a target located 84 pixels above and 156 pixels to the left of his position. Choose a polar coordinate system with the player at the pole and the polar axis extending to the player’s right. Find the polar coordinates of the target (this determines the distance and angle at which the player should fire his gun). Find r to the nearest pixel and in degree measure to the nearest tenth of a degree.92PEExplain the significance of r0 in an ordered pair r, in polar coordinates.Explain why a point in the plane can be represented by infinitely many ordered pairs in polar coordinates.95PEUse the results of Exercise 95. a. If 1=2 what is the distance between P and Q? Explain the significance of the answer. b. If 12=2 what is the distance between P and Q? Explain the significance of the answer.Given points Pr1,1 and Qr2,2 represented in polar coordinates with 12 and r11 and r20 , use the law of cosines to show that the distance d between P and Q is given byd=r12+r222r1r2cos21 .Use the results of Exercise 97. Find the distance between the points 2,6 and 4,3 .99PEFor Exercises 99-102, use a graphing utility to graph the polar equation for the given intervals of ,x, and y to confirm your answer to the given exercise. Use the window setting for 0,4.83x4.83 , and 3y3 . r=sec; Exercise 60For Exercises 99-102, use a graphing utility to graph the polar equation for the given intervals of ,x, and y to confirm your answer to the given exercise. Use the window setting for 0,4.83x4.83 , and 3y3 . r=4cos; Exercise 61For Exercises 99-102, use a graphing utility to graph the polar equation for the given intervals of ,x, and y to confirm your answer to the given exercise. Use the window setting for 0,4.83x4.83 , and 3y3 . r=2sin; Exercise 62Graph r=4cos .Graph r=3+3sin .Graph r=1+4cos .4SPGraph r2=25cos2 .Graph r=0.8 for 0 .1PEIf replacing r, by r, in a polar equation results in an equivalent equation, then the graph of the equation is symmetric with respect to the .3PE4PEDetermining the ordered pairs for which r=0 in a polar equation gives what information?6PE7PE8PE9PE10PE11PE12PE13PE14PE15PEFor Exercises 15-22, determine whether the tests for symmetry in Table 7-2 detect symmetry with respect to a. The polar axis. Replace r, by r, b. The line =2 . Replace r, by r, c. The pole. Replace r, by r, Otherwise, indicate that the test is inconclusive. r=5cos4For Exercises 15-22, determine whether the tests for symmetry in Table 7-2 detect symmetry with respect to a. The polar axis. Replace r, by r, b. The line =2 . Replace r, by r, c. The pole. Replace r, by r, Otherwise, indicate that the test is inconclusive. r=3cos3For Exercises 15-22, determine whether the tests for symmetry in Table 7-2 detect symmetry with respect to a. The polar axis. Replace r, by r, b. The line =2 . Replace r, by r, c. The pole. Replace r, by r, Otherwise, indicate that the test is inconclusive. r=64cosFor Exercises 15-22, determine whether the tests for symmetry in Table 7-2 detect symmetry with respect to a. The polar axis. Replace r, by r, b. The line =2 . Replace r, by r, c. The pole. Replace r, by r, Otherwise, indicate that the test is inconclusive. r=34+sinFor Exercises 15-22, determine whether the tests for symmetry in Table 7-2 detect symmetry with respect to a. The polar axis. Replace r, by r, b. The line =2 . Replace r, by r, c. The pole. Replace r, by r, Otherwise, indicate that the test is inconclusive. r=3+2sinFor Exercises 15-22, determine whether the tests for symmetry in Table 7-2 detect symmetry with respect to a. The polar axis. Replace r, by r, b. The line =2 . Replace r, by r, c. The pole. Replace r, by r, Otherwise, indicate that the test is inconclusive. r=2For Exercises 15-22, determine whether the tests for symmetry in Table 7-2 detect symmetry with respect to a. The polar axis. Replace r, by r, b. The line =2 . Replace r, by r, c. The pole. Replace r, by r, Otherwise, indicate that the test is inconclusive. r2=9sin2For Exercises 23-30, use symmetry to graph the polar curve and identify the type of curve. (See Examples 2-4) r=22sin24PEFor Exercises 23-30, use symmetry to graph the polar curve and identify the type of curve. (See Examples 2-4) r=24cosFor Exercises 23-30, use symmetry to graph the polar curve and identify the type of curve. (See Examples 2-4) r=1+7sinFor Exercises 23-30, use symmetry to graph the polar curve and identify the type of curve. (See Examples 2-4) r=3cos5For Exercises 23-30, use symmetry to graph the polar curve and identify the type of curve. (See Examples 2-4) r=8sin4For Exercises 23-30, use symmetry to graph the polar curve and identify the type of curve. (See Examples 2-4) r=32sin30PE31PE32PE33PE34PE35PE36PE37PE38PE39PE40PE41PE42PE43PE44PEFor Exercises 43-46, graph the polar curve. (See Example 5-6) r=0.5 for 0For Exercises 43-46, graph the polar curve. (See Example 5-6) r=0.6 for 0For Exercises 47-52, for the purpose of determining a suitable viewing window for the graph of the equation, a. Find the maximum value of r . b. Experiment with a graphing utility to find an interval for over which the graph is traced only once. r=6cosFor Exercises 47-52, for the purpose of determining a suitable viewing window for the graph of the equation, a. Find the maximum value of r . b. Experiment with a graphing utility to find an interval for over which the graph is traced only once. r=4sinFor Exercises 47-52, for the purpose of determining a suitable viewing window for the graph of the equation, a. Find the maximum value of r . b. Experiment with a graphing utility to find an interval for over which the graph is traced only once. r=3sinFor Exercises 47-52, for the purpose of determining a suitable viewing window for the graph of the equation, a. Find the maximum value of r .b. Experiment with a graphing utility to find an interval for over which the graph is traced only once. r=5+2cosFor Exercises 47-52, for the purpose of determining a suitable viewing window for the graph of the equation, a. Find the maximum value of r . b. Experiment with a graphing utility to find an interval for over which the graph is traced only once. r=3cos2For Exercises 47-52, for the purpose of determining a suitable viewing window for the graph of the equation, a. Find the maximum value of r . b. Experiment with a graphing utility to find an interval for over which the graph is traced only once. r=7sin453PE54PE55PE56PEA quilter wants to make a design in the shape of a daisy where the "petals" are 3.5 in. long. She plans to use a computer to print a graph of the outline of the figure. Determine an equation she could use.The outline of a propeller on an airplane is in the shape of a lemniscate. Write an equation that could be used for the outline.a. Given r=8sin4 , determine the set of values of for which r=0 on the interval [0,2) . b. Use a graphing utility to graph r=8sin4 on the given intervals. i. 02 ii. 2 iii. 32 iv. 322a. Given r=6sin3 , determine the set of values of for which r=0 on the interval 0, . b. Use a graphing utility to graph r=6sin3 on the given intervals. i. 03 ii. 323 iii. 23If r=fsin , that is, if r is expressed as a function of sin , what type of symmetry does the graph have? Explain.62PE63PE64PEFor Exercises 63-66, use the results of Exercises 61-62 to determine by inspection whether the graph of the given equation is symmetric with respect to the polar axis or to the line =2 . r=2cos23cos+1For Exercises 63-66, use the results of Exercises 61-62 to determine by inspection whether the graph of the given equation is symmetric with respect to the polar axis or to the line =2 . r=15cosGiven r=1+sec (conchoids), a. For what value(s) of on the interval 0,2 is r undefined? b. Use a graphing utility to graph r=1+sec on the interval [0,2) . c. Discuss the behavior of the graph for values of near those found in part (a).Given r=2 (hyperbolic spiral), a. For what value of is r undefined? b. Use a graphing utility to graph r=2 on the interval 4,4 . c. Discuss the behavior of the graph for values of near 0Use a graphing utility to graph the equations on the same viewing window for 02 . a. r=6sin2 b. r=6sin2+4 c. How do the graphs of r=6sin2 and r=6sin2+4 differ?Use a graphing utility to graph the equations on the same viewing window for 02 . a. r=2+4cos b. r=2+4cos4 c. r=2+4cos2 . d. Based on the results of parts (a)-(c), make a hypothesis about the effect of on the graph of r=f .A logarithmic spiral is represented by r=aeb . a. Graph r=0.5e0.1 and r=0.5e0.1 over the interval 04 . b. What appears to be the difference between the graphs for b=0.1 and b=0.1 ?An Archimedean spiral is represented by r=a . a. Graph r=0.5 and r=0.5 over the interval 08 and use a ZOOM square viewing window. b. Archimedean spirals have the property that a ray through the origin will intersect successive turns of the spiral at a constant distance of 2a . What is the distance between each point of the spiral r=0.5 along the line =2 ?Graph the limaqons given by the equation r=a+bcos for a=1,2,3,4,5, and 6, and b=3 . Comment on the effect of ab on the graph.Graph the limacons given by the equation r=a+bcos for a=1,2, and 3, and b=4 . Comment on the effect of ab on the graph.Find the values of on the interval 0,2 for which the graphs of r=1+sin and r=1sin intersect.Find the values of on the interval 0,2 for which the graphs of r=cos and r=1cos intersect.Given r=6cos2 , a. Replace r, by r, in the equation. Does this show that the graph of the equation is symmetric with respect to the line =2 ? b. The ordered pair r,2 is another representation of the point r, . Replace r, by r,2 in the equation. Does this show that the graph of the equationis symmetric with respect to the line =2 ? c. What other type of symmetry (if any) does the graph of the equation have? d. Use a graphing utility to graph the equation on a graphing utility for 04 .Given r=4sin4 , a. Replace r, by r, in the equation. Does this show that the graph of the equation is symmetric with respect to the polar axis? b. The ordered pair r,4 is another representation of the point r, . Replace r, by r,4 in the equation. Does this show that the graph of the equation is symmetric with respect to the polar axis? c. What other type of symmetry (if any) does the graph of the equation have? d. Use a graphing utility to graph the equation on a graphing utility for 08 .The graph of r=4cos2 is shown. a. Is 4,2 a solution to the equation? b. The point 4,2 , appears to lie on the curve. How do you explain this?80PEa. Match the polar or rectangular equation on the left with the letter of the rectangular or polar equation on the right. b. Describe the graph. c. Determine if the graph is symmetric with respect to the origin (pole), x-axis (polar axis), or y-axis line=2 . A. 2x+y=4 B. r=10sin C. r=12 D. =23 E. y=5 F. r=3sec G. x12+y12=2 H. x+32+y2=9 x2+y2=1442PRE3PREa. Match the polar or rectangular equation on the left with the letter of the rectangular or polar equation on the right. b. Describe the graph. c. Determine if the graph is symmetric with respect to the origin (pole), x-axis (polar axis), or y-axis line=2 . A. 2x+y=4 B. r=10sin C. r=12 D. =23 E. y=5 F. r=3sec G. x12+y12=2 H. x+32+y2=9 r=6cos5PRE6PRE7PREa. Match the polar or rectangular equation on the left with the letter of the rectangular or polar equation on the right. b. Describe the graph. c. Determine if the graph is symmetric with respect to the origin (pole), x-axis (polar axis), or y-axis line=2 . A. 2x+y=4 B. r=10sin C. r=12 D. =23 E. y=5 F. r=3sec G. x12+y12=2 H. x+32+y2=9 r=2sin+2cos9PRE10PRE11PRE12PRE13PREGraph the complex numbers in the complex plane. a. 1+3i b. 45i c. 3 d. 4iFind the modulus. a. z=512i b. z=4+6iWrite the complex number in polar form. a. z=1i b. z=3+4iWrite z=6cos150+isin150 in rectangular form.Given z1=3cos12+isin12 and z2=2cos6+isin6 , find z1z2 and write the product in polar form.Given z1=18cos75+isin75 and z2=2cos255+isin255 , find z1z2 and write the quotient in polar form.Compute 4cos70+isin703 and write the result in rectangular form a + bi.Compute 22i4 and write the result in rectangular form a+bi .Find the cube roots of 64 over the set of complex numbers. Write the results in rectangular form a+bi .Find the fourth roots of 3+33i over the set of complex numbers. Write the results in polar form using degree measure for the argument.The complex plane has two axes. The horizontal axis is called the axis, and the vertical axis is called the axis.2PEThe rectangular form of a complex number z is a+bi and the polar form (or trigonometric form) is given by z= , where r= and tan= for 0 .Given z=rcos+isin , the value r is called the of z and is called a(n) of z.If z1=r1cos1+isin1 and z2=r2cos2+isin2 , then z1z2= and z1z2= for z20 .If z=rcos+isin , then zn = for a positive integer n. An nth root of z is a complex number w such that =z .For Exercises 7-8, graph the complex numbers in the complex plane. (See Example 1) a. z=3+4i b. z=25i c. z=2i d. z=38PEFor Exercises 9-14, find the modulus of each complex number. (See Example 2) a. 2021i b. 11+60iFor Exercises 9-14, find the modulus of each complex number. (See Example 2) a. 5+12i b. 724i11PE12PE13PEFor Exercises 9-14, find the modulus of each complex number. (See Example 2) a. 222i b. 1+33i15PE16PE17PE18PE19PE20PEFor Exercises 15-22, write the complex number in polar form with 02 . (See Example 3) a. 17 b. 4i22PEFor Exercises 23-30, convert the complex number from polar form to rectangular form a+bi .(See Example 4) 18cos53+isin53For Exercises 23-30, convert the complex number from polar form to rectangular form a+bi .(See Example 4) 35cos330+isin330For Exercises 23-30, convert the complex number from polar form to rectangular form a+bi .(See Example 4) 15cos315+isin315For Exercises 23-30, convert the complex number from polar form to rectangular form a+bi .(See Example 4) 24cos6+isin6For Exercises 23-30, convert the complex number from polar form to rectangular form a+bi (See Example 4) 18.6cos43+isin4328PEFor Exercises 23-30, convert the complex number from polar form to rectangular form a+bi . (See Example 4) 43cos90+isin90For Exercises 23-30, convert the complex number from polar form to rectangular form a+bi .(See Example 4) 5cos180+isin18031PE32PEFor Exercises 31-42, given complex numbers z1 and z2 , a. Find z1z2 and write the product in polar form. b. Find z1z2 and write the quotient in polar form. (See Example 5-6) z1=3cos74+isin74,z2=6cos712+isin712For Exercises 31-42, given complex numbers z1 and z2 , a. Find z1z2 and write the product in polar form. b. Find z1z2 and write the quotient in polar form. (See Example 5-6) z1=10cos1112+isin1112,z2=2cos54+isin5435PE36PEFor Exercises 31-42, given complex numbers z1 and z2 , a. Find z1z2 and write the product in polar form. b. Find z1z2 and write the quotient in polar form. (See Example 5-6) z1=34cos12+isin12,z2=112cos512+isin512For Exercises 31-42, given complex numbers z1 and z2 , a. Find z1z2 and write the product in polar form. b. Find z1z2 and write the quotient in polar form. (See Example 5-6) z1=56cos4+isin4,z2=30cos34+isin34For Exercises 31-42, given complex numbers z1 and z2 , a. Find z1z2 and write the product in polar form. b. Find z1z2 and write the quotient in polar form. (See Example 5-6) z1=22i and z2=3+3i40PE41PEFor Exercises 31-42, given complex numbers z1 and z2 , a. Find z1z2 and write the product in polar form. b. Find z1z2 and write the quotient in polar form. (See Example 5-6) z1=24+243i,z2=434iFor Exercises 43-52, use De Moivre's theorem to find the indicated power. Write the result in rectangular form a+bi . (See Example 7-8) 6cos36+isin366For Exercises 43-52, use De Moivre's theorem to find the indicated power. Write the result in rectangular form a+bi . (See Example 7-8) 4cos415+isin415545PEFor Exercises 43-52, use De Moivre's theorem to find the indicated power. Write the result in rectangular form a+bi . (See Example 7-8) 3cos38+isin384For Exercises 43-52, use De Moivre's theorem to find the indicated power. Write the result in rectangular form a+bi (See Examples 7-8). 3cos35+isin359For Exercises 43-52, use De Moivre's theorem to find the indicated power. Write the result in rectangular form a+bi (See Examples 7-8). 5cos60+isin605For Exercises 43-52, use De Moivre's theorem to find the indicated power. Write the result in rectangular form a+bi (See Examples 7-8). 2cos26.25+isin26.258For Exercises 43-52, use De Moivre's theorem to find the indicated power. Write the result in rectangular form a+bi (See Examples 7-8). 4cos33.75+isin33.754For Exercises 43-52, use De Moivre's theorem to find the indicated power. Write the result in rectangular form a+bi (See Examples 7-8). 232i5For Exercises 43-52, use De Moivre's theorem to find the indicated power. Write the result in rectangular form a+bi (See Examples 7-8). 4+43i3When expressing the fourth roots wk of a complex number generated by wk =rncos+360kn+isin+360kn for k=0,1,2,,n1 by how many degrees will consecutive roots differ? In general, given an integer n2 , by how many degrees will consecutive nth roots differ?When expressing the fifth roots Wk of a complex number generated by wk =rncos+2kn+isin+2kn for k=0,1,2,...,n1 , by how many radians will consecutive roots differ?For Exercises 55-60, the indicated complex roots by first writing the number in polar form. Write the results in rectangular form a+bi . (See Examples 9) The sixth roots of 729For Exercises 55-60, the indicated complex roots by first writing the number in polar form. Write the results in rectangular form a+bi . (See Examples 9) The fourth roots of 625For Exercises 55-60, the indicated complex roots by first writing the number in polar form. Write the results in rectangular form a+bi . (See Examples 9) The cube roots of 27iFor Exercises 55-60, the indicated complex roots by first writing the number in polar form. Write the results in rectangular form a+bi . (See Examples 9) The cube roots of 125iFor Exercises 55-60, the indicated complex roots by first writing the number in polar form. Write the results in rectangular form a+bi . (See Examples 9) The square roots of 8+83iFor Exercises 55-60, the indicated complex roots by first writing the number in polar form. Write the results in rectangular form a+bi . (See Examples 9) The square roots of 50503iFor Exercises 61-64, find the indicated complex roots. Write the results in polar form. The square roots of 16cos34+isin34For Exercises 61-64, find the indicated complex roots. Write the results in polar form. The square roots of 49cos116+isin116For Exercises 61-64, find the indicated complex roots. Write the results in polar form. The fifth roots of 163+16iFor Exercises 61-64, find the indicated complex roots. Write the results in polar form. The sixth roots of 2222iFor Exercises 55-60, the indicated complex roots by first writing the number in polar form. Write the results in rectangular form a+bi . (See Examples 10) The cube roots of 27cos114+isin114For Exercises 55-60, the indicated complex roots by first writing the number in polar form. Write the results in rectangular form a+bi . (See Examples 10) The fourth roots of 256cos228+isin22867PE68PEFor Exercises 65-70, find the indicated complex roots. Write the results in polar form using degree measure for the argument. (See Example 10) The sixth roots of 223iFor Exercises 65-70, find the indicated complex roots. Write the results in polar form using degree measure for the argument. (See Example 10) The fifth roots of 535i .The square roots, cube roots, and fourth roots of + are shown in the figure from left to right, respectively. Use the pattern to describe how to find the nth roots of 1.Use the results of Exercise 71 to find the fifth roots of 1.73PEUse De Moivre's theorem to show that 1i is a cube root of 22cos54+isin54 .For Exercises 75-78, shade the area in the complex plane. z=a+bia3,b276PE77PE78PE79PE80PE81PEFor Exercises 79-82, convert the complex number from polar form to rectangular form, a+bi . cos285+isin28583PEGiven z1=2+2i and z2=3i , a. Find z1z2 by dividing the numbers in rectangular form and then converting the quotient to polar form. b. Find z1z2 by dividing the numbers in polar form.Let Z1=r1cos1+isin1 and Z2=r2cos2+sin2 .Prove thatz1z2=r1r2cos12+isin12z20 .Show that Z1=rcos+isin and Z2=rcos=isin are conjugates.Let z=rcos+isin . a.Prove that z1=r1cos+isin b. Prove that z2=r2cos2+isin2 Repeating this argument, we can show that De Moivre's theorem zn=rcos+isinn=rncosn+isinn also holds for negative integers.Use De Moivre's theorem to find the indicated power. Write the result in rectangular form a+bi . a.4242i2 b.1=3i4 c.3212i3For Exercises 89-92, simplify and write the solution in rectangular form, a+bi . 1+i31i4For Exercises 89-92, simplify and write the solution in rectangular form, a+bi . 1+i43i5For Exercises 89-92, simplify and write the solution in rectangular form, a+bi . 3+3i513i2For Exercises 89-92, simplify and write the solution in rectangular form, a+bi . 5252i43i3Solve x6=64 by factoring and applying the zero product rule to show that the results are the same as the six 6th roots of 64 as found in Example 9 on page720 .Solve x3=64 by factoring and applying the zero product rule to show that the results are the same as the three cube roots of 64 as found in Skill Practice Exercise 9 on page720 .95PEFor Exercises 95-98, find all complex solutions to the equation. Write the solutions in polar form. z5=1621+i97PEFor Exercises 95-98, find all complex solutions to the equation. Write the solutions in polar form. z3=8iHow is the absolute value of a complex number z=a+bi similar to the absolute value of a real number xi ?Is the polar representation of a complex number z=a+bi unique? Explain.Euler's formula states that ei=cos+isin . a. Evaluate e/4i . b. Use Euler's formula to show thatei+1=0 .This equation relates five fundamental numbers used in mathematics 0,1,e,,and,i.In calculus, we can show that ex=1+x1+x221+x3321+x44321+.... , cosx=1x221+x44321x6654321+...... , and sinx=xx3321+x554321x77654321+...... Write the first eight terms of the expansion of eix to illustrate that eix=cosx+isinx .Show that 3ei/12 is a solution to z881z4+6561=0 .Show that e7i/12 is a solution to z8z4+1=0.1SPSuppose that vector v has initial point 4,5 and terminal point1,12 . a. Find the component form of v. b. If v is placed with initial point at2,4 , what is the terminal point of v?3SPGiven a=11,7 and b=4,6 , find a. 6a b. 3a+10b5SP6SP7SPA ball is thrown from a height of 1m . At the time of release, the velocity of the ball is given byv=22.2i+12.7j , where each component is given in m/sec. Find the initial velocity v and the angle θ above the horizontal (in degrees) at which the ball was thrown. Round to 1 decimal place.A plane travels with an air speed of 400 knots at S20 E and encounters a wind of 50 knots acting in the direction of S30 w Rounding to the nearest whole unit, a. Express the velocity of the plane p relative to the air in terms of i and j. b. Express the velocity of the wind w in terms of i and j. c. Find the true speed (ground speed) and bearing of the plane.10SP11SP12SPA is a quantity that has magnitude only, whereas a vector has both magnitude andGiven a vector AB the initial point is and the terminal point is .The ofAB of is the length of the vector and is denoted by .For a real number k, the vector kv is in the same direction as v if k is and in the opposite direction as v if k is .Two vectors r=a1,b1 and s=a2,b2 are equal if .9PE10PEFor Exercises 11-16, vector v has initial point P and terminal point Q. Vector w has initial point R and terminal point S. (See Example 1) a. Find the magnitude of v. b. Find the magnitude of w. c. Determine whether v=w and explain your reasoning. P4,7,Q5,10andR2,8,S1,11For Exercises 11-16, vector v has initial point P and terminal point Q. Vector w has initial point R and terminal point S. (See Example 1) a. Find the magnitude of v. b. Find the magnitude of w. c. Determine whether v=w and explain your reasoning. P2,10,Q5,8andR9,3,S6,1For Exercises 11-16, vector v has initial point P and terminal point Q. Vector w has initial point R and terminal point S. (See Example 1) a. Find the magnitude of v. b. Find the magnitude of w. c. Determine whether v=w and explain your reasoning. P4,1,Q7,6andR5,7,S2,1214PEFor Exercises 11-16, vector v has initial point P and terminal point Q. Vector w has initial point R and terminal point S. (See Example 1) a. Find the magnitude of v. b. Find the magnitude of w. c. Determine whether v=w and explain your reasoning. P12,10,Q16,7andR9,3,S1,916PE17PE18PEFor Exercises 17-20, refer to vectors v and w in the figure. Sketch v and 2w in standard position.20PE21PE22PE23PE24PE25PE26PE27PEGiven vector v with initial point 17,80 and terminal point72,53 , a. Find the component form of v. b. If v is placed with initial point at13,12 , what is the terminal point of v?Given v=17,29 with initial point 4,10 , find the terminal point of v.Given v=5,8 with initial point1,7 , find the terminal point of v.For Exercises 31-46, perform the indicated operations for the given vectors. (See Examples 3-4) v=8,10w=3,7s=1,2r=2,10 v+wFor Exercises 31-46, perform the indicated operations for the given vectors. (See Examples 3-4) v=8,10w=3,7s=1,2r=2,10 s+wFor Exercises 31-46, perform the indicated operations for the given vectors. (See Examples 3-4) v=8,10w=3,7s=1,2r=2,10 v+rFor Exercises 31-46, perform the indicated operations for the given vectors. (See Examples 3-4) v=8,10w=3,7s=1,2r=2,10 r+sFor Exercises 31-46, perform the indicated operations for the given vectors. (See Examples 3-4) v=8,10w=3,7s=1,2r=2,10 wvFor Exercises 31-46, perform the indicated operations for the given vectors. (See Examples 3-4) v=8,10w=3,7s=1,2r=2,10 wsFor Exercises 31-46, perform the indicated operations for the given vectors. (See Examples 3-4) v=8,10w=3,7s=1,2r=2,10 rwFor Exercises 31-46, perform the indicated operations for the given vectors. (See Examples 3-4) v=8,10w=3,7s=1,2r=2,10 vr39PEFor Exercises 31-46, perform the indicated operations for the given vectors. (See Examples 3-4) v=8,10w=3,7s=1,2r=2,10 4wFor Exercises 31-46, perform the indicated operations for the given vectors. (See Examples 3-4) v=8,10w=3,7s=1,2r=2,10 12v42PE43PE44PEFor Exercises 31-46, perform the indicated operations for the given vectors. (See Examples 3-4) v=8,10w=3,7s=1,2r=2,10 2rw+sFor Exercises 31-46, perform the indicated operations for the given vectors. (See Examples 3-4) v=8,10w=3,7s=1,2r=2,10 3v+2wsFor Exercises 47-54, let c and d be scalars and let v=a.1,b1,w=a2,b2,u=a3,b3and0=0,0 . Prove the given statement. v+w+u=v+w+uFor Exercises 47-54, let c and d be scalars and let v=a.1,b1,w=a2,b2,u=a3,b3and0=0,0 . Prove the given statement. v+0=0+v