{"id":2367,"date":"2025-08-13T00:52:46","date_gmt":"2025-08-13T00:52:46","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=2367"},"modified":"2026-02-18T20:07:49","modified_gmt":"2026-02-18T20:07:49","slug":"parametric-functions-and-vectors-get-stronger","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/parametric-functions-and-vectors-get-stronger\/","title":{"raw":"Parametric Functions and Vectors: Get Stronger","rendered":"Parametric Functions and Vectors: Get Stronger"},"content":{"raw":"

Parametric Equations<\/h1>\r\n1. What is a system of parametric equations?\r\n\r\n3. Explain how to eliminate a parameter given a set of parametric equations.\r\n\r\nFor the following exercises, eliminate the parameter [latex]t[\/latex] to rewrite the parametric equation as a Cartesian equation.\r\n\r\n7. [latex]\\begin{cases}x\\left(t\\right)=5-t\\hfill \\\\ y\\left(t\\right)=8 - 2t\\hfill \\end{cases}[\/latex]\r\n\r\n9. [latex]\\begin{cases}x\\left(t\\right)=2t+1\\hfill \\\\ y\\left(t\\right)=3\\sqrt{t}\\hfill \\end{cases}[\/latex]\r\n\r\n11. [latex]\\begin{cases}x\\left(t\\right)=2{e}^{t}\\hfill \\\\ y\\left(t\\right)=1 - 5t\\hfill \\end{cases}[\/latex]\r\n\r\n13. [latex]\\begin{cases}x\\left(t\\right)=4\\text{log}\\left(t\\right)\\hfill \\\\ y\\left(t\\right)=3+2t\\hfill \\end{cases}[\/latex]\r\n\r\n15. [latex]\\begin{cases}x\\left(t\\right)={t}^{3}-t\\hfill \\\\ y\\left(t\\right)=2t\\hfill \\end{cases}[\/latex]\r\n\r\n17. [latex]\\begin{cases}x\\left(t\\right)={e}^{2t}\\hfill \\\\ y\\left(t\\right)={e}^{6t}\\hfill \\end{cases}[\/latex]\r\n\r\n19. [latex]\\begin{cases}x\\left(t\\right)=4\\text{cos}t\\hfill \\\\ y\\left(t\\right)=5\\sin t \\hfill \\end{cases}[\/latex]\r\n\r\n21. [latex]\\begin{cases}x\\left(t\\right)=2{\\text{cos}}^{2}t\\hfill \\\\ y\\left(t\\right)=-\\sin t \\hfill \\end{cases}[\/latex]\r\n\r\n23. [latex]\\begin{cases}x\\left(t\\right)=t - 1\\\\ y\\left(t\\right)={t}^{2}\\end{cases}[\/latex]\r\n\r\n25. [latex]\\begin{cases}x\\left(t\\right)=2t - 1\\\\ y\\left(t\\right)={t}^{3}-2\\end{cases}[\/latex]\r\n\r\nFor the following exercises, rewrite the parametric equation as a Cartesian equation by building an [latex]x\\text{-}y[\/latex] table.\r\n\r\n27. [latex]\\begin{cases}x\\left(t\\right)=4-t\\\\ y\\left(t\\right)=3t+2\\end{cases}[\/latex]\r\n\r\n29. [latex]\\begin{cases}x\\left(t\\right)=4t - 1\\\\ y\\left(t\\right)=4t+2\\end{cases}[\/latex]\r\n\r\nFor the following exercises, parameterize (write parametric equations for) each Cartesian equation by setting [latex]x\\left(t\\right)=t[\/latex] or by setting [latex]y\\left(t\\right)=t[\/latex].\r\n\r\n31. [latex]y\\left(x\\right)=2\\sin x+1[\/latex]\r\n\r\n33. [latex]x\\left(y\\right)=\\sqrt{y}+2y[\/latex]\r\n\r\nFor the following exercises, parameterize (write parametric equations for) each Cartesian equation by using [latex]x\\left(t\\right)=a\\cos t[\/latex] and [latex]y\\left(t\\right)=b\\sin t[\/latex]. Identify the curve.\r\n\r\n35. [latex]\\frac{{x}^{2}}{16}+\\frac{{y}^{2}}{36}=1[\/latex]\r\n\r\n37. [latex]{x}^{2}+{y}^{2}=10[\/latex]\r\n\r\n39. Parameterize the line from [latex]\\left(-1,0\\right)[\/latex] to [latex]\\left(3,-2\\right)[\/latex] so that the line is at [latex]\\left(-1,0\\right)[\/latex] at [latex]t=0[\/latex], and at [latex]\\left(3,-2\\right)[\/latex] at [latex]t=1[\/latex].\r\n\r\n41. Parameterize the line from [latex]\\left(4,1\\right)[\/latex] to [latex]\\left(6,-2\\right)[\/latex] so that the line is at [latex]\\left(4,1\\right)[\/latex] at [latex]t=0[\/latex], and at [latex]\\left(6,-2\\right)[\/latex] at [latex]t=1[\/latex].\r\n\r\nFor the following exercises, use a graphing calculator to complete the table of values for each set of parametric equations.\r\n\r\n45. [latex]\\begin{cases}{x}_{1}\\left(t\\right)={t}^{2}-4\\hfill \\\\ {y}_{1}\\left(t\\right)=2{t}^{2}-1\\hfill \\end{cases}[\/latex]<\/span>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]t[\/latex]<\/th>\r\n[latex]x[\/latex]<\/th>\r\n[latex]y[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
1<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
2<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
3<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n47. Find two different sets of parametric equations for [latex]y={\\left(x+1\\right)}^{2}[\/latex].<\/span>\r\n\r\n49. Find two different sets of parametric equations for [latex]y={x}^{2}-4x+4[\/latex].\r\n

Parametric Equations: Graphs<\/h1>\r\n1. What are two methods used to graph parametric equations?\r\n\r\n3. Why are some graphs drawn with arrows?\r\n\r\n5. Why are parametric graphs important in understanding projectile motion?\r\n\r\nFor the following exercises, graph each set of parametric equations by making a table of values. Include the orientation on the graph.\r\n\r\n7. [latex]\\begin{cases}x\\left(t\\right)=t - 1\\hfill \\\\ y\\left(t\\right)={t}^{2}\\hfill \\end{cases}[\/latex]<\/span>\r\n\r\n\r\n\r\n\r\n\r\n
[latex]t[\/latex] <\/strong><\/td>\r\n[latex]-3[\/latex]<\/td>\r\n[latex]-2[\/latex]<\/td>\r\n[latex]-1[\/latex]<\/td>\r\n[latex]0[\/latex]<\/td>\r\n[latex]1[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]x[\/latex] <\/strong><\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
[latex]y[\/latex] <\/strong><\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n9. [latex]\\begin{cases}x\\left(t\\right)=-2 - 2t\\hfill \\\\ y\\left(t\\right)=3+t\\hfill \\end{cases}[\/latex]\r\n\r\n\r\n\r\n\r\n\r\n
[latex]t[\/latex] <\/strong><\/td>\r\n[latex]-3[\/latex]<\/td>\r\n[latex]-2[\/latex]<\/td>\r\n[latex]-1[\/latex]<\/td>\r\n[latex]0[\/latex]<\/td>\r\n[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]x[\/latex] <\/strong><\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
[latex]y[\/latex] <\/strong><\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n11. [latex]\\begin{cases}x\\left(t\\right)={t}^{2}\\hfill \\\\ y\\left(t\\right)=t+3\\hfill \\end{cases}[\/latex]\r\n\r\n\r\n\r\n\r\n\r\n
[latex]t[\/latex] <\/strong><\/td>\r\n[latex]-2[\/latex]<\/td>\r\n[latex]-1[\/latex]<\/td>\r\n[latex]0[\/latex]<\/td>\r\n[latex]1[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]x[\/latex] <\/strong><\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
[latex]y[\/latex] <\/strong><\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFor the following exercises, sketch the curve and include the orientation.\r\n\r\n13. [latex]\\begin{cases}x\\left(t\\right)=-\\sqrt{t}\\\\ y\\left(t\\right)=t\\end{cases}[\/latex]\r\n\r\n15. [latex]\\begin{cases}x\\left(t\\right)=-t+2\\\\ y\\left(t\\right)=5-|t|\\end{cases}[\/latex]\r\n\r\n17. [latex]\\begin{cases}x\\left(t\\right)=2\\text{sin}t\\hfill \\\\ y\\left(t\\right)=4\\text{cos}t\\hfill \\end{cases}[\/latex]\r\n\r\n19. [latex]\\begin{cases}x\\left(t\\right)=3{\\cos }^{2}t\\\\ y\\left(t\\right)=-3{\\sin }^{2}t\\end{cases}[\/latex]\r\n\r\n21. [latex]\\begin{cases}x\\left(t\\right)=\\sec t\\\\ y\\left(t\\right)={\\tan }^{2}t\\end{cases}[\/latex]\r\n\r\nFor the following exercises, graph the equation and include the orientation. Then, write the Cartesian equation.\r\n\r\n23. [latex]\\begin{cases}x\\left(t\\right)=t - 1\\hfill \\\\ y\\left(t\\right)=-{t}^{2}\\hfill \\end{cases}[\/latex]\r\n\r\n25. [latex]\\begin{cases}x\\left(t\\right)=2\\cos t\\\\ y\\left(t\\right)=-\\sin t\\end{cases}[\/latex]\r\n\r\n27. [latex]\\begin{cases}x\\left(t\\right)={e}^{2t}\\\\ y\\left(t\\right)=-{e}^{t}\\end{cases}[\/latex]\r\n\r\nFor the following exercises, graph the equation and include the orientation.\r\n\r\n29. [latex]x=2t,y={t}^{2},-5\\le t\\le 5[\/latex]\r\n\r\n31. [latex]x\\left(t\\right)=-t,y\\left(t\\right)=\\sqrt{t},t\\ge 0[\/latex]\r\n\r\n41. If the parametric equations [latex]x\\left(t\\right)={t}^{2}[\/latex] and [latex]y\\left(t\\right)=6 - 3t[\/latex] have the graph of a horizontal parabola opening to the right, what would change the direction of the curve?\r\n\r\n63.\u00a0An object is thrown in the air with vertical velocity of 20 ft\/s and horizontal velocity of 15 ft\/s. The object\u2019s height can be described by the equation [latex]y\\left(t\\right)=-16{t}^{2}+20t[\/latex] , while the object moves horizontally with constant velocity 15 ft\/s. Write parametric equations for the object\u2019s position, and then eliminate time to write height as a function of horizontal position.\r\n\r\nFor the following exercises, use this scenario: A dart is thrown upward with an initial velocity of 65 ft\/s at an angle of elevation of 52\u00b0. Consider the position of the dart at any time [latex]t[\/latex]. Neglect air resistance.\r\n\r\n65. Find parametric equations that model the problem situation.\r\n\r\n67. When will the dart hit the ground?\r\n\r\n69. At what time will the dart reach maximum height?\r\n

Vectors<\/h1>\r\n3.\u00a0What are i<\/em><\/strong> and j<\/em><\/strong>, and what do they represent?\r\n\r\n4.\u00a0What is component form?\r\n\r\n5.\u00a0When a unit vector is expressed as [latex]\\langle a,b\\rangle[\/latex],\u00a0which letter is the coefficient of the i<\/em><\/strong> and which the j<\/em><\/strong>?\r\n\r\n7.\u00a0Given a vector with initial point (\u22124,2) and terminal point (3,\u22123), find an equivalent vector whose initial point is (0,0). Write the vector in component form [latex]\\langle a,b\\rangle[\/latex].\r\n\r\nFor the following exercises, determine whether the two vectors u<\/em><\/strong> and v<\/em><\/strong> are equal, where u<\/strong><\/em> has an initial point [latex]P_{1}[\/latex] and a terminal point [latex]P_{2}[\/latex] and v<\/em><\/strong> has an initial point [latex]P3[\/latex] and a terminal point [latex]P4[\/latex].\r\n\r\n9. [latex]P_{1}=\\left(5,1\\right)\\text{, }P_{2}=\\left(3,\u22122\\right),P_{3}=\\left(\u22121,3\\right)[\/latex], and [latex]P_{4}=\\left(9,\u22124\\right)[\/latex]\r\n\r\n11. [latex]P_{1}=\\left(\u22121,\u22121\\right),P_{2}=\\left(\u22124,5\\right),P_{3}=\\left(\u221210,6\\right)[\/latex], and [latex]P_{4}=\\left(\u221213,12\\right)[\/latex]\r\n\r\n13. [latex]P_{1}=\\left(8,3\\right),P_{2}=\\left(6,5\\right),P_{3}=\\left(11,8\\right)[\/latex], and [latex]P_{4}=\\left(9,10\\right)[\/latex]\r\n\r\n15.\u00a0Given initial point [latex]P_{1}=\\left(6,0\\right)[\/latex] and terminal point [latex]P_{2}=\\left(\u22121,\u22123\\right)[\/latex], write the vector v<\/em> in terms of i<\/em> and j.\r\n\r\nFor the following exercises, use the vectors [latex]\\boldsymbol{u}=\\boldsymbol{i}+5\\boldsymbol{j}[\/latex], [latex]\\boldsymbol{v}=\u22122\\boldsymbol{i}\u22123\\boldsymbol{j}[\/latex], and [latex]\\boldsymbol{w}=4\\boldsymbol{i}\u2212\\boldsymbol{j}[\/latex].\r\n\r\n17. Find [latex]4\\boldsymbol{v}+2\\boldsymbol{u}[\/latex]\r\n\r\nFor the following exercises, use the given vectors to compute [latex]\\boldsymbol{u}+\\boldsymbol{v}[\/latex], [latex]\\boldsymbol{u}\u2212\\boldsymbol{v}[\/latex], and [latex]2\\boldsymbol{u}\u22123\\boldsymbol{v}[\/latex].\r\n\r\n19. [latex]\\boldsymbol{u}=\\langle\u22123,4\\rangle,\\boldsymbol{v}=\\langle\u22122,1\\rangle[\/latex]\r\n\r\n21.\u00a0Let [latex]\\boldsymbol{v}=5\\boldsymbol{i}+2\\boldsymbol{j}[\/latex]. Find a vector that is twice the length and points in the opposite direction as v<\/strong><\/em>.\r\n\r\nFor the following exercises, find a unit vector in the same direction as the given vector.\r\n\r\n23. [latex]\\boldsymbol{b}=\u22122\\boldsymbol{i}+5\\boldsymbol{j}[\/latex]\r\n\r\n25. [latex]\\boldsymbol{d}=\u2013\\frac{1}{3}\\boldsymbol{i}+\\frac{5}{2}\\boldsymbol{j}[\/latex]\r\n\r\n27.\u00a0[latex]\\boldsymbol{u}=\u201314\\boldsymbol{i}+2\\boldsymbol{j}[\/latex]\r\n\r\nFor the following exercises, find the magnitude and direction of the vector, [latex]0\\leq\\theta<2\\pi[\/latex].\r\n\r\n29.\u00a0[latex]\\langle 6,5\\rangle[\/latex]\r\n\r\n31.\u00a0[latex]\\langle \u20134,\u20136\\rangle[\/latex]\r\n\r\n33.\u00a0Given [latex]\\boldsymbol{u}=\u2212\\boldsymbol{i}\u2212\\boldsymbol{j}[\/latex] and [latex]\\boldsymbol{v}=\\boldsymbol{i}+5\\boldsymbol{j}[\/latex], calculate [latex]\\boldsymbol{u}\\cdot \\boldsymbol{v}[\/latex].\r\n\r\n35.\u00a0Given [latex]\\boldsymbol{u}=\\langle\u22121,6\\rangle[\/latex] and [latex]\\boldsymbol{v}=\\langle 6,\u22121\\rangle[\/latex], calculate [latex]\\boldsymbol{u}\\cdot \\boldsymbol{v}[\/latex].\r\n\r\nFor the following exercises, given v<\/em><\/strong>, draw v<\/em><\/strong>, 3v<\/em><\/strong>,\u00a0and [latex]\\frac{1}{2}\\boldsymbol{v}[\/latex].\r\n\r\n37.\u00a0[latex]\\langle \u22121,4\\rangle[\/latex]\r\n\r\nFor the following exercises, use the vectors shown to sketch [latex]\\boldsymbol{u}+\\boldsymbol{v}[\/latex], [latex]\\boldsymbol{u}\u2212\\boldsymbol{v}[\/latex], and [latex]2\\boldsymbol{u}[\/latex].\r\n\r\n39.\r\n\"Plot\r\n\r\n41.\r\n\"Plot\r\n\r\nFor the following exercises, use the vectors shown to sketch [latex]2\\boldsymbol{u}+\\boldsymbol{v}[\/latex].\r\n\r\n43.\r\n\"Plot\r\n\r\nFor the following exercises, use the vectors shown to sketch [latex]\\boldsymbol{u}\u22123\\boldsymbol{v}[\/latex].\r\n\r\n45.\r\n\"Plot\r\n\r\nFor the following exercises, write the vector shown in component form.\r\n\r\n47.\r\n\"Insert\r\n\r\n49.\u00a0Given initial point [latex]P_{1}=\\left(4,\u22121\\right)[\/latex] and terminal point [latex]P_{2}=\\left(\u22123,2\\right)[\/latex], write the vector v<\/em><\/strong> in terms of i<\/em><\/strong> and j<\/em><\/strong>. Draw the points and the vector on the graph.\r\n\r\nFor the following exercises, use the given magnitude and direction in standard position, write the vector in component form.\r\n\r\n51. [latex]|\\boldsymbol{v}|=6,\\theta=45^{\\circ}[\/latex]\r\n\r\n53.\u00a0[latex]|\\boldsymbol{v}|=2,\\theta=300^{\\circ}[\/latex]\r\n\r\n55.\u00a0A 60-pound box is resting on a ramp that is inclined 12\u00b0. Rounding to the nearest tenth,\r\n
Find the magnitude of the normal (perpendicular) component of the force.\r\nFind the magnitude of the component of the force that is parallel to the ramp.<\/div>\r\n57.\u00a0Find the magnitude of the horizontal and vertical components of a vector with magnitude 8 pounds pointed in a direction of [latex]27^{\\circ}[\/latex]\u00a0above the horizontal. Round to the nearest hundredth.\r\n\r\n59.\u00a0Find the magnitude of the horizontal and vertical components of the vector with magnitude 5 pounds pointed in a direction of [latex]55^{\\circ}[\/latex] above the horizontal. Round to the nearest hundredth.\r\n\r\n61.\u00a0A woman leaves home and walks 3 miles west, then 2 miles southwest. How far from home is she, and in what direction must she walk to head directly home?\r\n\r\n63.\u00a0A man starts walking from home and walks 4 miles east, 2 miles southeast, 5 miles south, 4 miles southwest, and 2 miles east. How far has he walked? If he walked straight home, how far would he have to walk?\r\n\r\n65.\u00a0A man starts walking from home and walks 3 miles at [latex]20^{\\circ}[\/latex] north of west, then 5 miles at [latex]10^{\\circ}[\/latex] west of south, then 4 miles at [latex]15^{\\circ}[\/latex] north of east. If he walked straight home, how far would he have to the walk, and in what direction?\r\n\r\n67.\u00a0An airplane is heading north at an airspeed of 600 km\/hr, but there is a wind blowing from the southwest at 80 km\/hr. How many degrees off course will the plane end up flying, and what is the plane\u2019s speed relative to the ground?\r\n\r\n69.\u00a0An airplane needs to head due north, but there is a wind blowing from the southwest at 60 km\/hr. The plane flies with an airspeed of 550 km\/hr. To end up flying due north, how many degrees west of north will the pilot need to fly the plane?\r\n\r\n71.\u00a0As part of a video game, the point (5,7) is rotated counterclockwise about the origin through an angle of [latex]35^{\\circ}[\/latex]. Find the new coordinates of this point.\r\n\r\n73.\u00a0Two children are throwing a ball back and forth straight across the back seat of a car. The ball is being thrown 10 mph relative to the car, and the car is traveling 25 mph down the road. If one child doesn't catch the ball, and it flies out the window, in what direction does the ball fly (ignoring wind resistance)?\r\n\r\n75.\u00a0A 50-pound object rests on a ramp that is inclined 19\u00b0. Find the magnitude of the components of the force parallel to and perpendicular to (normal) the ramp to the nearest tenth of a pound.\r\n\r\n77.\u00a0Suppose a body has a force of 10 pounds acting on it to the right, 25 pounds acting on it \u2500135\u00b0 from the horizontal, and 5 pounds acting on it directed 150\u00b0 from the horizontal. What single force is the resultant force acting on the body?\r\n\r\n79.\u00a0Suppose a body has a force of 3 pounds acting on it to the left, 4 pounds acting on it upward, and 2 pounds acting on it [latex]30^{\\circ}[\/latex] from the horizontal. What single force is needed to produce a state of equilibrium on the body? Draw the vector.","rendered":"

Parametric Equations<\/h1>\n

1. What is a system of parametric equations?<\/p>\n

3. Explain how to eliminate a parameter given a set of parametric equations.<\/p>\n

For the following exercises, eliminate the parameter [latex]t[\/latex] to rewrite the parametric equation as a Cartesian equation.<\/p>\n

7. [latex]\\begin{cases}x\\left(t\\right)=5-t\\hfill \\\\ y\\left(t\\right)=8 - 2t\\hfill \\end{cases}[\/latex]<\/p>\n

9. [latex]\\begin{cases}x\\left(t\\right)=2t+1\\hfill \\\\ y\\left(t\\right)=3\\sqrt{t}\\hfill \\end{cases}[\/latex]<\/p>\n

11. [latex]\\begin{cases}x\\left(t\\right)=2{e}^{t}\\hfill \\\\ y\\left(t\\right)=1 - 5t\\hfill \\end{cases}[\/latex]<\/p>\n

13. [latex]\\begin{cases}x\\left(t\\right)=4\\text{log}\\left(t\\right)\\hfill \\\\ y\\left(t\\right)=3+2t\\hfill \\end{cases}[\/latex]<\/p>\n

15. [latex]\\begin{cases}x\\left(t\\right)={t}^{3}-t\\hfill \\\\ y\\left(t\\right)=2t\\hfill \\end{cases}[\/latex]<\/p>\n

17. [latex]\\begin{cases}x\\left(t\\right)={e}^{2t}\\hfill \\\\ y\\left(t\\right)={e}^{6t}\\hfill \\end{cases}[\/latex]<\/p>\n

19. [latex]\\begin{cases}x\\left(t\\right)=4\\text{cos}t\\hfill \\\\ y\\left(t\\right)=5\\sin t \\hfill \\end{cases}[\/latex]<\/p>\n

21. [latex]\\begin{cases}x\\left(t\\right)=2{\\text{cos}}^{2}t\\hfill \\\\ y\\left(t\\right)=-\\sin t \\hfill \\end{cases}[\/latex]<\/p>\n

23. [latex]\\begin{cases}x\\left(t\\right)=t - 1\\\\ y\\left(t\\right)={t}^{2}\\end{cases}[\/latex]<\/p>\n

25. [latex]\\begin{cases}x\\left(t\\right)=2t - 1\\\\ y\\left(t\\right)={t}^{3}-2\\end{cases}[\/latex]<\/p>\n

For the following exercises, rewrite the parametric equation as a Cartesian equation by building an [latex]x\\text{-}y[\/latex] table.<\/p>\n

27. [latex]\\begin{cases}x\\left(t\\right)=4-t\\\\ y\\left(t\\right)=3t+2\\end{cases}[\/latex]<\/p>\n

29. [latex]\\begin{cases}x\\left(t\\right)=4t - 1\\\\ y\\left(t\\right)=4t+2\\end{cases}[\/latex]<\/p>\n

For the following exercises, parameterize (write parametric equations for) each Cartesian equation by setting [latex]x\\left(t\\right)=t[\/latex] or by setting [latex]y\\left(t\\right)=t[\/latex].<\/p>\n

31. [latex]y\\left(x\\right)=2\\sin x+1[\/latex]<\/p>\n

33. [latex]x\\left(y\\right)=\\sqrt{y}+2y[\/latex]<\/p>\n

For the following exercises, parameterize (write parametric equations for) each Cartesian equation by using [latex]x\\left(t\\right)=a\\cos t[\/latex] and [latex]y\\left(t\\right)=b\\sin t[\/latex]. Identify the curve.<\/p>\n

35. [latex]\\frac{{x}^{2}}{16}+\\frac{{y}^{2}}{36}=1[\/latex]<\/p>\n

37. [latex]{x}^{2}+{y}^{2}=10[\/latex]<\/p>\n

39. Parameterize the line from [latex]\\left(-1,0\\right)[\/latex] to [latex]\\left(3,-2\\right)[\/latex] so that the line is at [latex]\\left(-1,0\\right)[\/latex] at [latex]t=0[\/latex], and at [latex]\\left(3,-2\\right)[\/latex] at [latex]t=1[\/latex].<\/p>\n

41. Parameterize the line from [latex]\\left(4,1\\right)[\/latex] to [latex]\\left(6,-2\\right)[\/latex] so that the line is at [latex]\\left(4,1\\right)[\/latex] at [latex]t=0[\/latex], and at [latex]\\left(6,-2\\right)[\/latex] at [latex]t=1[\/latex].<\/p>\n

For the following exercises, use a graphing calculator to complete the table of values for each set of parametric equations.<\/p>\n

45. [latex]\\begin{cases}{x}_{1}\\left(t\\right)={t}^{2}-4\\hfill \\\\ {y}_{1}\\left(t\\right)=2{t}^{2}-1\\hfill \\end{cases}[\/latex]<\/span><\/p>\n\n\n\n\n\n\n\n
[latex]t[\/latex]<\/th>\n[latex]x[\/latex]<\/th>\n[latex]y[\/latex]<\/th>\n<\/tr>\n<\/thead>\n
1<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
2<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
3<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

47. Find two different sets of parametric equations for [latex]y={\\left(x+1\\right)}^{2}[\/latex].<\/span><\/p>\n

49. Find two different sets of parametric equations for [latex]y={x}^{2}-4x+4[\/latex].<\/p>\n

Parametric Equations: Graphs<\/h1>\n

1. What are two methods used to graph parametric equations?<\/p>\n

3. Why are some graphs drawn with arrows?<\/p>\n

5. Why are parametric graphs important in understanding projectile motion?<\/p>\n

For the following exercises, graph each set of parametric equations by making a table of values. Include the orientation on the graph.<\/p>\n

7. [latex]\\begin{cases}x\\left(t\\right)=t - 1\\hfill \\\\ y\\left(t\\right)={t}^{2}\\hfill \\end{cases}[\/latex]<\/span><\/p>\n\n\n\n\n\n
[latex]t[\/latex] <\/strong><\/td>\n[latex]-3[\/latex]<\/td>\n[latex]-2[\/latex]<\/td>\n[latex]-1[\/latex]<\/td>\n[latex]0[\/latex]<\/td>\n[latex]1[\/latex]<\/td>\n[latex]2[\/latex]<\/td>\n<\/tr>\n
[latex]x[\/latex] <\/strong><\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
[latex]y[\/latex] <\/strong><\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

9. [latex]\\begin{cases}x\\left(t\\right)=-2 - 2t\\hfill \\\\ y\\left(t\\right)=3+t\\hfill \\end{cases}[\/latex]<\/p>\n\n\n\n\n\n
[latex]t[\/latex] <\/strong><\/td>\n[latex]-3[\/latex]<\/td>\n[latex]-2[\/latex]<\/td>\n[latex]-1[\/latex]<\/td>\n[latex]0[\/latex]<\/td>\n[latex]1[\/latex]<\/td>\n<\/tr>\n
[latex]x[\/latex] <\/strong><\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
[latex]y[\/latex] <\/strong><\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

11. [latex]\\begin{cases}x\\left(t\\right)={t}^{2}\\hfill \\\\ y\\left(t\\right)=t+3\\hfill \\end{cases}[\/latex]<\/p>\n\n\n\n\n\n
[latex]t[\/latex] <\/strong><\/td>\n[latex]-2[\/latex]<\/td>\n[latex]-1[\/latex]<\/td>\n[latex]0[\/latex]<\/td>\n[latex]1[\/latex]<\/td>\n[latex]2[\/latex]<\/td>\n<\/tr>\n
[latex]x[\/latex] <\/strong><\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
[latex]y[\/latex] <\/strong><\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

For the following exercises, sketch the curve and include the orientation.<\/p>\n

13. [latex]\\begin{cases}x\\left(t\\right)=-\\sqrt{t}\\\\ y\\left(t\\right)=t\\end{cases}[\/latex]<\/p>\n

15. [latex]\\begin{cases}x\\left(t\\right)=-t+2\\\\ y\\left(t\\right)=5-|t|\\end{cases}[\/latex]<\/p>\n

17. [latex]\\begin{cases}x\\left(t\\right)=2\\text{sin}t\\hfill \\\\ y\\left(t\\right)=4\\text{cos}t\\hfill \\end{cases}[\/latex]<\/p>\n

19. [latex]\\begin{cases}x\\left(t\\right)=3{\\cos }^{2}t\\\\ y\\left(t\\right)=-3{\\sin }^{2}t\\end{cases}[\/latex]<\/p>\n

21. [latex]\\begin{cases}x\\left(t\\right)=\\sec t\\\\ y\\left(t\\right)={\\tan }^{2}t\\end{cases}[\/latex]<\/p>\n

For the following exercises, graph the equation and include the orientation. Then, write the Cartesian equation.<\/p>\n

23. [latex]\\begin{cases}x\\left(t\\right)=t - 1\\hfill \\\\ y\\left(t\\right)=-{t}^{2}\\hfill \\end{cases}[\/latex]<\/p>\n

25. [latex]\\begin{cases}x\\left(t\\right)=2\\cos t\\\\ y\\left(t\\right)=-\\sin t\\end{cases}[\/latex]<\/p>\n

27. [latex]\\begin{cases}x\\left(t\\right)={e}^{2t}\\\\ y\\left(t\\right)=-{e}^{t}\\end{cases}[\/latex]<\/p>\n

For the following exercises, graph the equation and include the orientation.<\/p>\n

29. [latex]x=2t,y={t}^{2},-5\\le t\\le 5[\/latex]<\/p>\n

31. [latex]x\\left(t\\right)=-t,y\\left(t\\right)=\\sqrt{t},t\\ge 0[\/latex]<\/p>\n

41. If the parametric equations [latex]x\\left(t\\right)={t}^{2}[\/latex] and [latex]y\\left(t\\right)=6 - 3t[\/latex] have the graph of a horizontal parabola opening to the right, what would change the direction of the curve?<\/p>\n

63.\u00a0An object is thrown in the air with vertical velocity of 20 ft\/s and horizontal velocity of 15 ft\/s. The object\u2019s height can be described by the equation [latex]y\\left(t\\right)=-16{t}^{2}+20t[\/latex] , while the object moves horizontally with constant velocity 15 ft\/s. Write parametric equations for the object\u2019s position, and then eliminate time to write height as a function of horizontal position.<\/p>\n

For the following exercises, use this scenario: A dart is thrown upward with an initial velocity of 65 ft\/s at an angle of elevation of 52\u00b0. Consider the position of the dart at any time [latex]t[\/latex]. Neglect air resistance.<\/p>\n

65. Find parametric equations that model the problem situation.<\/p>\n

67. When will the dart hit the ground?<\/p>\n

69. At what time will the dart reach maximum height?<\/p>\n

Vectors<\/h1>\n

3.\u00a0What are i<\/em><\/strong> and j<\/em><\/strong>, and what do they represent?<\/p>\n

4.\u00a0What is component form?<\/p>\n

5.\u00a0When a unit vector is expressed as [latex]\\langle a,b\\rangle[\/latex],\u00a0which letter is the coefficient of the i<\/em><\/strong> and which the j<\/em><\/strong>?<\/p>\n

7.\u00a0Given a vector with initial point (\u22124,2) and terminal point (3,\u22123), find an equivalent vector whose initial point is (0,0). Write the vector in component form [latex]\\langle a,b\\rangle[\/latex].<\/p>\n

For the following exercises, determine whether the two vectors u<\/em><\/strong> and v<\/em><\/strong> are equal, where u<\/strong><\/em> has an initial point [latex]P_{1}[\/latex] and a terminal point [latex]P_{2}[\/latex] and v<\/em><\/strong> has an initial point [latex]P3[\/latex] and a terminal point [latex]P4[\/latex].<\/p>\n

9. [latex]P_{1}=\\left(5,1\\right)\\text{, }P_{2}=\\left(3,\u22122\\right),P_{3}=\\left(\u22121,3\\right)[\/latex], and [latex]P_{4}=\\left(9,\u22124\\right)[\/latex]<\/p>\n

11. [latex]P_{1}=\\left(\u22121,\u22121\\right),P_{2}=\\left(\u22124,5\\right),P_{3}=\\left(\u221210,6\\right)[\/latex], and [latex]P_{4}=\\left(\u221213,12\\right)[\/latex]<\/p>\n

13. [latex]P_{1}=\\left(8,3\\right),P_{2}=\\left(6,5\\right),P_{3}=\\left(11,8\\right)[\/latex], and [latex]P_{4}=\\left(9,10\\right)[\/latex]<\/p>\n

15.\u00a0Given initial point [latex]P_{1}=\\left(6,0\\right)[\/latex] and terminal point [latex]P_{2}=\\left(\u22121,\u22123\\right)[\/latex], write the vector v<\/em> in terms of i<\/em> and j.<\/p>\n

For the following exercises, use the vectors [latex]\\boldsymbol{u}=\\boldsymbol{i}+5\\boldsymbol{j}[\/latex], [latex]\\boldsymbol{v}=\u22122\\boldsymbol{i}\u22123\\boldsymbol{j}[\/latex], and [latex]\\boldsymbol{w}=4\\boldsymbol{i}\u2212\\boldsymbol{j}[\/latex].<\/p>\n

17. Find [latex]4\\boldsymbol{v}+2\\boldsymbol{u}[\/latex]<\/p>\n

For the following exercises, use the given vectors to compute [latex]\\boldsymbol{u}+\\boldsymbol{v}[\/latex], [latex]\\boldsymbol{u}\u2212\\boldsymbol{v}[\/latex], and [latex]2\\boldsymbol{u}\u22123\\boldsymbol{v}[\/latex].<\/p>\n

19. [latex]\\boldsymbol{u}=\\langle\u22123,4\\rangle,\\boldsymbol{v}=\\langle\u22122,1\\rangle[\/latex]<\/p>\n

21.\u00a0Let [latex]\\boldsymbol{v}=5\\boldsymbol{i}+2\\boldsymbol{j}[\/latex]. Find a vector that is twice the length and points in the opposite direction as v<\/strong><\/em>.<\/p>\n

For the following exercises, find a unit vector in the same direction as the given vector.<\/p>\n

23. [latex]\\boldsymbol{b}=\u22122\\boldsymbol{i}+5\\boldsymbol{j}[\/latex]<\/p>\n

25. [latex]\\boldsymbol{d}=\u2013\\frac{1}{3}\\boldsymbol{i}+\\frac{5}{2}\\boldsymbol{j}[\/latex]<\/p>\n

27.\u00a0[latex]\\boldsymbol{u}=\u201314\\boldsymbol{i}+2\\boldsymbol{j}[\/latex]<\/p>\n

For the following exercises, find the magnitude and direction of the vector, [latex]0\\leq\\theta<2\\pi[\/latex].\n\n29.\u00a0[latex]\\langle 6,5\\rangle[\/latex]\n\n31.\u00a0[latex]\\langle \u20134,\u20136\\rangle[\/latex]\n\n33.\u00a0Given [latex]\\boldsymbol{u}=\u2212\\boldsymbol{i}\u2212\\boldsymbol{j}[\/latex] and [latex]\\boldsymbol{v}=\\boldsymbol{i}+5\\boldsymbol{j}[\/latex], calculate [latex]\\boldsymbol{u}\\cdot \\boldsymbol{v}[\/latex].\n\n35.\u00a0Given [latex]\\boldsymbol{u}=\\langle\u22121,6\\rangle[\/latex] and [latex]\\boldsymbol{v}=\\langle 6,\u22121\\rangle[\/latex], calculate [latex]\\boldsymbol{u}\\cdot \\boldsymbol{v}[\/latex].\n\nFor the following exercises, given v<\/em><\/strong>, draw v<\/em><\/strong>, 3v<\/em><\/strong>,\u00a0and [latex]\\frac{1}{2}\\boldsymbol{v}[\/latex].<\/p>\n

37.\u00a0[latex]\\langle \u22121,4\\rangle[\/latex]<\/p>\n

For the following exercises, use the vectors shown to sketch [latex]\\boldsymbol{u}+\\boldsymbol{v}[\/latex], [latex]\\boldsymbol{u}\u2212\\boldsymbol{v}[\/latex], and [latex]2\\boldsymbol{u}[\/latex].<\/p>\n

39.
\n\"Plot<\/p>\n

41.
\n\"Plot<\/p>\n

For the following exercises, use the vectors shown to sketch [latex]2\\boldsymbol{u}+\\boldsymbol{v}[\/latex].<\/p>\n

43.
\n\"Plot<\/p>\n

For the following exercises, use the vectors shown to sketch [latex]\\boldsymbol{u}\u22123\\boldsymbol{v}[\/latex].<\/p>\n

45.
\n\"Plot<\/p>\n

For the following exercises, write the vector shown in component form.<\/p>\n

47.
\n\"Insert<\/p>\n

49.\u00a0Given initial point [latex]P_{1}=\\left(4,\u22121\\right)[\/latex] and terminal point [latex]P_{2}=\\left(\u22123,2\\right)[\/latex], write the vector v<\/em><\/strong> in terms of i<\/em><\/strong> and j<\/em><\/strong>. Draw the points and the vector on the graph.<\/p>\n

For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.<\/p>\n

51. [latex]|\\boldsymbol{v}|=6,\\theta=45^{\\circ}[\/latex]<\/p>\n

53.\u00a0[latex]|\\boldsymbol{v}|=2,\\theta=300^{\\circ}[\/latex]<\/p>\n

55.\u00a0A 60-pound box is resting on a ramp that is inclined 12\u00b0. Rounding to the nearest tenth,<\/p>\n

Find the magnitude of the normal (perpendicular) component of the force.
\nFind the magnitude of the component of the force that is parallel to the ramp.<\/div>\n

57.\u00a0Find the magnitude of the horizontal and vertical components of a vector with magnitude 8 pounds pointed in a direction of [latex]27^{\\circ}[\/latex]\u00a0above the horizontal. Round to the nearest hundredth.<\/p>\n

59.\u00a0Find the magnitude of the horizontal and vertical components of the vector with magnitude 5 pounds pointed in a direction of [latex]55^{\\circ}[\/latex] above the horizontal. Round to the nearest hundredth.<\/p>\n

61.\u00a0A woman leaves home and walks 3 miles west, then 2 miles southwest. How far from home is she, and in what direction must she walk to head directly home?<\/p>\n

63.\u00a0A man starts walking from home and walks 4 miles east, 2 miles southeast, 5 miles south, 4 miles southwest, and 2 miles east. How far has he walked? If he walked straight home, how far would he have to walk?<\/p>\n

65.\u00a0A man starts walking from home and walks 3 miles at [latex]20^{\\circ}[\/latex] north of west, then 5 miles at [latex]10^{\\circ}[\/latex] west of south, then 4 miles at [latex]15^{\\circ}[\/latex] north of east. If he walked straight home, how far would he have to the walk, and in what direction?<\/p>\n

67.\u00a0An airplane is heading north at an airspeed of 600 km\/hr, but there is a wind blowing from the southwest at 80 km\/hr. How many degrees off course will the plane end up flying, and what is the plane\u2019s speed relative to the ground?<\/p>\n

69.\u00a0An airplane needs to head due north, but there is a wind blowing from the southwest at 60 km\/hr. The plane flies with an airspeed of 550 km\/hr. To end up flying due north, how many degrees west of north will the pilot need to fly the plane?<\/p>\n

71.\u00a0As part of a video game, the point (5,7) is rotated counterclockwise about the origin through an angle of [latex]35^{\\circ}[\/latex]. Find the new coordinates of this point.<\/p>\n

73.\u00a0Two children are throwing a ball back and forth straight across the back seat of a car. The ball is being thrown 10 mph relative to the car, and the car is traveling 25 mph down the road. If one child doesn’t catch the ball, and it flies out the window, in what direction does the ball fly (ignoring wind resistance)?<\/p>\n

75.\u00a0A 50-pound object rests on a ramp that is inclined 19\u00b0. Find the magnitude of the components of the force parallel to and perpendicular to (normal) the ramp to the nearest tenth of a pound.<\/p>\n

77.\u00a0Suppose a body has a force of 10 pounds acting on it to the right, 25 pounds acting on it \u2500135\u00b0 from the horizontal, and 5 pounds acting on it directed 150\u00b0 from the horizontal. What single force is the resultant force acting on the body?<\/p>\n

79.\u00a0Suppose a body has a force of 3 pounds acting on it to the left, 4 pounds acting on it upward, and 2 pounds acting on it [latex]30^{\\circ}[\/latex] from the horizontal. What single force is needed to produce a state of equilibrium on the body? Draw the vector.<\/p>\n","protected":false},"author":67,"menu_order":25,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":520,"module-header":"practice","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2367"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":6,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2367\/revisions"}],"predecessor-version":[{"id":5719,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2367\/revisions\/5719"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/520"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2367\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2367"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2367"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2367"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2367"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}