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Pre-AP Precalculus Module 10 Vectors and Parametric Equations 10.1 Vectors A vector is a quantity that has both magnitude (measurement) and direction. Itโs represented by a directed line segment with an arrowhead on one end, where the length of the segment represents magnitude and the arrowhead represents direction. If an arrowhead is applied to point ๐ on ฬ ฬ ฬ ฬ ๐๐ , then the โโโโโ has initial point ๐ and terminal point ๐. If vector ๐ has the same magnitude as vector ๐๐ โโโโโ , then vector ๐๐ we say ๐ = โโโโโ ๐๐. Two vectors ๐ and ๐ are equal if they have the same magnitude and direction regardless of the position of their initial points. ๐ = โ๐ if vectors ๐ and ๐ have the same magnitude but opposite directions. Note that in a typed format, when using a single letter to denote the name of a vector, we use a boldface letter, but in handwritten work, we do draw an arrow above the letter. When adding vectors together, we position each vector so that its initial point coincides with the terminal point of the preceding vector. The first vector can be placed anywhere. The sum (or resultant) is the vector that starts at the first initial point and ends at the final terminal point. Properties ๏ท ๏ท ๏ท ๏ท ๏ท ๏ท ๏ท ๏ท Commutative Property: ๐ + ๐ = ๐ + ๐ Associative Property: ๐ + (๐ + ๐) = (๐ + ๐) + ๐ ๐+๐ =๐+๐=๐ ๐ + (โ๐) = ๐ If ๐ผ is a scalar and ๐ is a vector, the scalar multiple ๐ผ๐ is defined as: o If ๐ผ > 0, ๐ผ๐ is the vector whose magnitude is ๐ผ times the magnitude of ๐ and whose direction is the same as ๐. o If ๐ผ < 0, ๐ผ๐ is the vector whose magnitude is |๐ผ| times the magnitude of ๐ and whose direction is opposite that of ๐. o If ๐ผ = 0 or if ๐ = ๐, then ๐ผ๐ = ๐. (๐ผ + ๐ฝ)๐ = ๐ผ๐ + ๐ฝ๐ ๐ผ(๐ + ๐) = ๐ผ๐ + ๐ผ๐ ๐ผ(๐ฝ๐) = (๐ผ๐ฝ)๐ Example 1 โ Use the given vectors to graph each of the following vectors: (a) ๐ โ ๐, (b) 2๐ + 3๐, and (c) 2๐ โ ๐ + ๐. Sullivan & Sullivan โ Section 9.4 10.1 Vectors We use the symbol โ๐โ to represent the magnitude of a vector ๐. Since โ๐โ equals the length of a directed line segment, ๏ท ๏ท ๏ท ๏ท โ๐โ โฅ 0 โ๐โ = 0 if and only if ๐ = ๐ โ๐โ = โโ๐โ โ๐ผ๐โ = |๐ผ|โ๐โ A vector ๐ for which โ๐โ = 1 is called a unit vector, and is denoted as ๐ขฬ. ๐ฬ = โฉ1,0โช is the unit vector in the direction of the positive ๐ฅ-axis, and ๐ฬ = โฉ0,1โช is the unit vector in the direction of the positive ๐ฆ-axis. If ๐ > 0 and ๐ > 0 are both scalar values, then ๏ท ๏ท ๏ท ๏ท ๐๐ฬ is a vector with magnitude ๐ in the direction of ๐ฬ ๐๐ฬ is the vector with magnitude ๐ in the direction of ๐ฬ โ๐๐ฬ is the vector with magnitude ๐ in the opposite direction of ๐ฬ โ๐๐ฬ is the vector with magnitude ๐ in the direction of ๐ฬ ๐ = ๐๐ฬ + ๐๐ฬ = โฉ๐, ๐โช is the component form of the resultant vector that extends ๐ units horizontally (to the left if ๐ < 0 and to the right if ๐ > 0) and ๐ units vertically (up if ๐ > 0 or down if ๐ < 0). ๐ and ๐ are the components of ๐. A vector that has an initial point at the origin is called a position vector, and as a consequence the terminal point of ๐ = โฉ๐, ๐โช is (๐, ๐). A vector whose initial point is not at the origin is called a displacement vector. Suppose that ๐ is a displacement vector with initial point ๐1 = (๐ฅ1 , ๐ฆ1 ), not necessarily the origin, and terminal point ๐2 = (๐ฅ2 , ๐ฆ2 ). If ๐ = โโโโโโโโ ๐1 ๐2 , then ๐ is equal to the position vector โฉ๐ฅ2 โ ๐ฅ1 , ๐ฆ2 โ ๐ฆ1 โช. Example 2 โ Find the displacement vector โโโโโโโโ ๐1 ๐2 if ๐1 = (โ1,2) and ๐2 = (4,6). If ๐ = โฉ๐1 , ๐1 โช and ๐ = โฉ๐2 , ๐2 โช, then: ๏ท ๏ท ๏ท ๐ = ๐ if and only if ๐1 = ๐2 and ๐1 = ๐2 . ๐ ± ๐ = โฉ๐1 ± ๐2 , ๐1 ± ๐2 โช = (๐1 ± ๐2 )๐ฬ + (๐1 ± ๐2 )๐ฬ ๐ผ๐ = โฉ๐ผ๐1 , ๐ผ๐1 โช = ๐ผ๐1 ๐ฬ + ๐ผ๐1 ๐ฬ Sullivan & Sullivan โ Section 9.4 10.1 Vectors Just like with polar complex notation, if ๐ = โฉ๐, ๐โช, then โ๐โ = โ๐2 + ๐ 2 and the direction of ๐ is ๐๐ฃ = ๐ ๐ tanโ1 ๐ or ๐๐ฃ = tanโ1 ๐ + 180°, depending on which quadrant the corresponding position vector would terminate in. Likewise, a vector ๐ with magnitude ๐ in the direction ๐๐ค can be written as ๐ = โฉ๐ cos ๐๐ค , ๐ sin ๐๐ค โช. Example 3 โ If ๐ = โฉ2,3โช and ๐ = โฉ3, โ4โช, find (a) ๐ + ๐ and (b) ๐ โ ๐. Example 4 โ If ๐ = โฉ2,3โช and ๐ = โฉ3, โ4โช, find (a) 3๐, (b) 2๐ โ 3๐, and (c) โ๐โ. ๐ For any nonzero vector ๐, the vector ๐ฃฬ = โ๐โ is a unit vector that has the same direction as ๐. As a consequence, ๐ = โ๐โ๐ฃฬ. Example 5 โ Find a unit vector in the same direction as ๐ = โฉ4, โ3โช. Sullivan & Sullivan โ Section 9.4 10.1 Vectors Example 6 โ A ball is thrown with an initial speed of 25 mph in a direction that makes an angle of 30° with the positive ๐ฅ-axis. Express the velocity vector ๐ in terms of ๐ฬ and ๐ฬ. What is the initial speed in the horizontal direction? What is the initial speed in the vertical direction? Example 7 โ Find the direction angles of (a) ๐ = โฉ4, โ4โช and (b) ๐ = โฉโ3, โ3โ3โช. Example 8 โ A Boeing 737 aircraft maintains a constant airspeed of 500 mph headed due south. The jet stream is 80 mph in the northeasterly direction (N45°E). (a) Express the velocity ๐๐ of the 737 relative to the air and the velocity ๐๐ค of the jet stream in terms of ๐ฬ and ๐ฬ. (b) Find the velocity of the 737 relative to the ground. (c) Find the actual speed and direction of the 737 relative to the ground. Sullivan & Sullivan โ Section 9.4 10.1 Vectors Example 9 โ Two movers require a magnitude of force of 300 pounds to push a piano up a ramp inclined at an angle 20° from the horizontal. How much does the piano weigh? An object is said to be in static equilibrium if the object is at rest and the sum of all forces acting on the object is zero. Example 10 โ A box of supplies that weighs 1200 pounds is suspended by two cables attached to the ceiling, one forming an angle of 30° with the ceiling and the other forming an angle of 45°. What are the tensions in the two cables? Complete the following exercises on a separate sheet of paper. In exercises 1 โ 8, use the figure to determine whether the given statement is true or false. 1. ๐จ + ๐ฉ = ๐ญ 2. ๐ฒ + ๐ฎ = ๐ญ B A F G H 3. ๐ช = ๐ซ โ ๐ฌ + ๐ญ E 4. ๐ฎ + ๐ฏ + ๐ฌ = ๐ซ 5. ๐ฌ + ๐ซ = ๐ฎ + ๐ฏ 6. ๐ฏ โ ๐ช = ๐ฎ โ ๐ญ 7. ๐จ + ๐ฉ + ๐ฒ + ๐ฎ = ๐ 8. ๐จ + ๐ฉ + ๐ช + ๐ฏ + ๐ฎ = ๐ Sullivan & Sullivan โ Section 9.4 C K D 10.1 Vectors In exercises 9 โ 12, find the displacement vector โโโโโโ ๐ท๐ธ in the form โฉ๐, ๐โช. 9. ๐ = (0,0); ๐ = (โ3, โ5) 11. ๐ = (โ1,4); ๐ = (6,2) 10. ๐ = (โ3,2); ๐ = (6,5) 12. ๐ = (1,1); ๐ = (2,2) In exercises 13 โ 18, find โ๐โ. 13. ๐ = 3๐ฬ โ 4๐ฬ 16. ๐ = โ๐ฬ โ ๐ฬฬ 14. ๐ = โ5๐ฬ + 12๐ฬ 17. ๐ = โ2๐ฬ + 3๐ฬ 15. ๐ = ๐ฬ โ ๐ฬฬ 18. ๐ = 6๐ฬ + 2๐ฬฬ In exercises 19 โ 24, find each quantity if ๐ = ๐๐ฬ โ ๐๐ฬ and ๐ = โ๐๐ฬ + ๐๐ฬ. 19. 2๐ + 3๐ 22. โ๐ + ๐โ 20. 3๐ โ 2๐ 23. โ๐โ โ โ๐โ 21. โ๐ โ ๐โ 24. โ๐โ + โ๐โ In exercises 25 โ 28, find the unit vector in the same direction as ๐. 25. ๐ = 3๐ฬ โ 4๐ฬ 27. ๐ = ๐ฬ โ ๐ฬฬ 26. ๐ = โ5๐ฬ + 12๐ฬ 28. ๐ = 2๐ฬ โ ๐ฬฬ 29. Find a vector v whose magnitude is 4 and whose component in the ๐ฬ direction is twice the component in the ๐ฬ direction. 30. Find a vector v whose magnitude is 3 and whose component in the ๐ฬ direction is equal to the component in the ๐ฬ direction. 31. If ๐ = 2๐ฬ โ ๐ฬ and ๐ = ๐ฅ๐ฬ + 3๐ฬ, find all numbers ๐ฅ for which โ๐ + ๐โ = 5. โโโโโ has length 5. 32. If ๐ = (โ3,1) and ๐ = (๐ฅ, 4), find all numbers ๐ฅ such that the vector represented by ๐๐ In exercises 33 โ 36, write the vector ๐ in the form โฉ๐, ๐โช, given its magnitude โ๐โ and the angle ๐ถ it makes with the positive ๐-axis. 33. โ๐โ = 8, ๐ผ = 45° 35. โ๐โ = 3; ๐ผ = 240° 34. โ๐โ = 14; ๐ผ = 120° 36. โ๐โ = 15; ๐ผ = 315° In exercises 37 โ 40, find the magnitude and direction angle of ๐ for each vector. 37. ๐ = ๐ฬ + โ3๐ฬฬ 39. ๐ = 6๐ฬ โ 4๐ฬฬ 38. ๐ = โ5๐ฬ โ 5๐ฬฬ 40. ๐ = โ๐ฬ + 3๐ฬฬ 41. Tw forces ๐ญ๐ and ๐ญ๐ of magnitudes 30 newtons (N) and 70 N act on an object at angles of 45° and 120°, respectfully. Find the direction and magnitude of the resultant force; i.e. ๐ญ๐ + ๐ญ๐ . Sullivan & Sullivan โ Section 9.4 10.1 Vectors 42. An Airbus A320 jet maintains a constant airspeed of 500 mph headed due north. The jet stream is 100 mph in the northeasterly direction (N45°E). a. Express the velocity ๐๐ of the A320 relative to the air and the velocity ๐๐ค of the jet stream in terms of ๐ฬ and ๐ฬ. b. Find the velocity of the A320 relative to the ground. c. Find the actual speed and direction of the A320 relative to the ground. 43. An airplane has an airspeed of 600 km/hr bearing S30°E. The wind velocity is 40 km/hr in the direction S45°E. Find the resultant vector representing the path of the plane relative to the ground. What is the ground speed of the plane? What is its direction? 44. A magnitude of 1200 pounds of force is required to prevent a car from rolling down a hill whose incline is 15° to the horizontal. What is the weight of the car? 45. The pilot of an aircraft wishes to head direction east but is faced with a wind speed of 40 mph from the northwest. If the pilot maintains an airspeed of 250 mph, what compass heading should be maintained to head directly east? What is the actual speed of the aircraft? 46. A weight of 800 pounds is suspended from two cables, which makes angles of 35° and 50° with the ceiling. What are the tensions in the two cables? 47. At a county fair truck pull, two pickup trucks are attached to the back end of a monster truck. One of the pickups pulls with a force of 2000 pounds and the other pulls with a force of 3000 pounds with an angle of 45° between them. With how much force must the monster truck pull in order to remain unmoved? (Hint: Find the resultant force of the two trucks.) 48. A farmer wishes to remove a stump from a field by pulling it out with his tractor. Having removed many stumps before, he estimates that he will need 6 tons (12,000 pounds) of force to remove the stump. However, his tractor is only capable of pulling with a force of 7000 pounds, so he asks his neighbor to help. His neighborโs tractor can pull with a force of 5500 pounds. They attach the two tractors to the stump with a 40° angle between the forces. a. Assuming the farmerโs estimate of a needed 6-ton force is correct, will the farmer be successful in removing the stump? Explain. b. Had the farmer arranged the tractors with a 25° angle between the forces, would he have been successful in removing the stump? Explain. Sullivan & Sullivan โ Section 9.4 10.2 The Dot Product If ๐ฃ = ๐1 ๐ฬ + ๐1 ๐ฬ and ๐ค โโ = ๐2 ๐ฬ + ๐2 ๐ฬ, are two vectors, then the dot or scalar product ๐ฃ โ ๐ค โโ = ๐1 ๐2 + ๐1 ๐2 . Example 1 โ If ๐ฃ = โฉ2, โ3โช and ๐ค โโ = โฉ5,3โช, find: (a) ๐ โ ๐ (b) ๐ โ ๐, (c) ๐ โ ๐, (d) ๐ โ ๐, (e) โ๐โ, and (f) โ๐โ. If u, v, and w are vectors, then ๏ท ๏ท ๏ท ๏ท ๐โ๐=๐โ๐ ๐ โ (๐ + ๐) = ๐ โ ๐ + ๐ โ ๐ ๐ โ ๐ = โ๐โ๐ ๐โ๐=๐ If u and v are two nonzero vectors, the angle ๐, 0 โค ๐ โค ๐, between u and v is determined by the formula ๐ โ ๐ = โ๐โโ๐โ cos ๐. Unlike matrix addition that required the vectors be positioned head-to-tail, matrix multiplication requires the vectors be placed tail-to-tail. Example 2 โ Find the angle ๐ between ๐ = 4๐ฬ โ 3๐ฬ and ๐ = 2๐ฬ + 5๐ฬ. Example 3 โ Find the angle ๐ between ๐ = โฉ3, โ1โช and ๐ = โฉ6, โ2โช. Sullivan & Sullivan โ Section 9.5 Foerster โ Section 10.4 10.2 The Dot Product Two vectors v and w are orthogonal if and only if ๐ โ ๐ = 0. Example 4 โ Show that ๐ = โฉ2, โ1โช and ๐ = โฉ3,6โช are orthogonal. In the last section, we decomposed vectors into their horizontal and vertical components. In fact, there are times when we will wish to decompose vectors into orthogonal components not aligned to the horizontal and vertical. For instance, looking back at Example 9 in 10.1, movers were moving a piano up the ramp. We decomposed the weight of the piano (a force vector point straight down) into the force vector perpendicular to the rampโs surface (friction) and the force vector running parallel to the rampโs surface. The component of a vector v in the direction of a second vector w is known as the vector projection. The magnitude of the vector projection is known as the scalar projection. If v and w have an angle ๐ between them when placed tail-to-tail, then the scalar projection of v onto w (or in the direction of w) is โ๐โ = โ๐โ cos ๐. For the vector projection, we must apply a direction to the scalar projection without changing the magnitude. Since the direction is determined by w, we can multiply the scalar projection by the unit ๐ vector in the direction of w, namely โ๐โ. Thus the vector projection ๐ = โ๐โ ๐โ๐โ cos ๐ . โ๐โ The numerator almost ๐โ๐ looks like ๐ โ ๐ = โ๐โโ๐โ cos ๐, and by multiplying by โ๐โ, we get the equivalent formula ๐ = โ๐โ2 ๐. The component orthogonal to the vector projection is equal to ๐ โ ๐. Example 5 โ Find the vector projection of ๐ = โฉ1,3โช onto ๐ = โฉ1,1โช. Decompose v into two vectors, ๐๐ and ๐๐ , where ๐๐ is parallel to w and ๐๐ is orthogonal to w. Sullivan & Sullivan โ Section 9.5 Foerster โ Section 10.4 10.2 The Dot Product Example 6 โ A wagon with two small children as occupants that weighs 100 pounds is on a hill with a grade of 20°. What is the magnitude of the force that is required to keep the wagon from rooling down the hill? In elementary physics, the work ๐ done by a constant force ๐น in moving an object from a point ๐ด to a โโโโโ โ. This definition presupposes point ๐ต is defined as ๐ = (๐๐๐๐๐๐ก๐ข๐๐ ๐๐ ๐๐๐๐๐)(๐๐๐ ๐ก๐๐๐๐) = โ๐ญโโ๐ด๐ต that the force is being done in the same direction as the motion, but if the force is applied at an angle ๐ to โโโโโ . the direction of the motion, then ๐ = ๐ญ โ ๐ด๐ต Example 7 โ A girl pulls a wagon along level ground with a force of 50 pounds. How much work is done in moving the wagon 100 feet if the handle makes an angle of 30° with the horizontal? Complete the following exercises on a separate sheet of paper. โ โ๐ โ and ๐ In exercises 1 โ 16, (a) find the dot product ๐ โโโ ; (b) find the angle between ๐ โโโ ; and (c) state whether the vectors are parallel, orthogonal, or neither. 1. ๐ฃ = โฉ1, โ1โช, ๐ค โโ = โฉ1,1โช 6. ๐ฃ = โฉ1, โ3โช, ๐ค โโ = โฉ1, โ1โช 2. ๐ฃ = โฉ1,1โช, ๐ค โโ = โฉโ1,1โช 7. ๐ฃ = โฉ3,4โช, ๐ค โโ = โฉโ6, โ8โช 3. ๐ฃ = โฉ2,1โช, ๐ค โโ = โฉ1, โ2โช 8. ๐ฃ = โฉ3, โ4โช, ๐ค โโ = โฉ9, โ12โช 4. ๐ฃ = โฉ2,2โช, ๐ค โโ = โฉ1,2โช 9. ๐ฃ = 4๐ฬ, ๐ค โโ = ๐ฬ 5. ๐ฃ = โฉโ3, โ1โช, ๐ค โโ = โฉ1,1โช 10. ๐ฃ = ๐ฬ, ๐ค โโ = โ3๐ฬ 11. Find ๐ so that the vectors โฉ1, โ๐โช and โฉ2,3โช are orthogonal. Sullivan & Sullivan โ Section 9.5 Foerster โ Section 10.4 10.2 The Dot Product 12. Find ๐ so that the vectors โฉ1,1โช and โฉ1, ๐โช are orthogonal. โ into two vectors โโโโ In exercises 13 โ 18, decompose ๐ ๐๐ and โโโโ ๐๐ , where โโโโ ๐๐ is parallel to ๐ โโโ and โโโโ ๐๐ is orthogonal to ๐ โโโ . 13. ๐ฃ = โฉ2, โ3โช, ๐ค โโ = โฉ1, โ1โช 16. ๐ฃ = โฉ2, โ1โช, ๐ค โโ = โฉ1, โ2โช 14. ๐ฃ = โฉโ3,2โช, ๐ค โโ = โฉ2,1โช 17. ๐ฃ = โฉ3,1โช, ๐ค โโ = โฉโ2, โ1โช 15. ๐ฃ = โฉ1, โ1โช, ๐ค โโ = โฉโ1, โ2โช 18. ๐ฃ = โฉ1, โ3โช, ๐ค โโ = โฉ4, โ1โช 19. Find the work done by a force of 3 pounds acting in the direction 60° to the horizontal in moving an object 6 feet from (0,0) to (6,0). 20. A wagon is pulled horizontally by exerting a force of 20 pounds on the handle at an angle of 30° with the horizontal. How much work is done in moving the wagon 100 feet? 21. The amount of energy collected by a solar panel depends on the intensity of the sunโs rays and the area of the panel. Let the vector I represent the intensity, in watts per square centimeter, having the direction of the sunโs rays. Let the vector A represent the area, in square centimeters, whose direction is the orientation of a solar panel. The total number of watts collected by the panel is given by ๐ = |๐ฐ โ ๐จ|. Suppose that ๐ฐ = โฉโ0.02, โ0.01โช and ๐จ = โฉ300,400โช. a. Find โ๐ฐโ and โ๐จโ and interpret the meaning of each. b. Compute ๐ and interpret its meaning. c. If the solar panel is to collect the maximum number of watts, what must be true about I and A? 22. Let the vector R represent the amount of rainfall, in inches, whose direction is the inclination of the rain to a rain gauge. Let the vector A represent the area, in square inches, whose direction is the orientation of the opening of the rain gauge. The volume of rain collected in the gauge, in cubic inches, is given by ๐ = |๐น โ ๐จ|, even when the rain falls in a slanted direction or the gauge is not perfectly vertical. Suppose that ๐น = โฉ0.75, โ1.75โช and ๐จ = โฉ0.3,1โช. a. Find โ๐นโ and โ๐จโ and interpret the meaning of each. b. Compute ๐ and interpret its meaning. c. If the gauge is to collect the maximum volume of rain, what must be true about R and A? 23. A Toyota Sienna with a gross weight of 5300 pounds is parked on a street with an 8° grade. Find the magnitude of the force required to keep the Sienna from rolling down the hill. What is the magnitude of the force perpendicular to he hill? 24. A Pontiac Bonneville with a gross weight of 4500 pounds is parked on a street with a 10° grade. Find the magnitude of the force required to keep the Bonneville from rolling down the hill. What is the magnitude of the force perpendicular to the hill? 25. Billy any Timmy are using a ramp to load furniture into a truck. While rolling a 250-pound piano up the ramp, they discover that the truck is too full of other furniture for the piano to fit. Timmy holds the piano in place on the ramp while Billy repositions other items to make room for it in the truck. If the angle of inclination of the ramp is 20°, how many pounds of force must Timmy exert to hold the piano in position? Sullivan & Sullivan โ Section 9.5 Foerster โ Section 10.4 10.2 The Dot Product 26. A bulldozer exerts 1000 pounds of force to prevent a 5000-pound boulder from rolling down a hill. Determine the angle of inclination of the hill. Sullivan & Sullivan โ Section 9.5 Foerster โ Section 10.4 10.3 Vectors in Space Coordinates in three dimensional Cartesian space are of the form (๐ฅ, ๐ฆ, ๐ง), where ๐ฅ = 0 is the equation for the ๐ฆ๐ง-plane, ๐ฆ = 0 is the ๐ฅ๐ง-plane, and ๐ง = 0 is the ๐ฅ๐ฆ-plane. If ๐1 = (๐ฅ1 , ๐ฆ1 , ๐ง1 ) and ๐2 = (๐ฅ2 , ๐ฆ2 , ๐ง2 ) are two points in space, the distance ๐ from ๐1 to ๐2 is ๐ = โ(๐ฅ2 โ ๐ฅ1 )2 + (๐ฆ2 โ ๐ฆ1 )2 + (๐ง2 โ ๐ง1 )2 . Example 1 โ Find the distance from ๐1 = (โ1,3,2) to ๐2 = (4, โ2,5). To represent vectors in space, we use unit vectors ๐ฬ, ๐ฬ and ๐ฬ, whose directions are along the positive ๐ฅaxis, positive ๐ฆ-axis, and positive ๐ง-axis, respectively. If a position vector has a terminal point at (๐, ๐, ๐), we can defined it as ๐๐ฬ + ๐๐ฬ + ๐๐ฬ = โฉ๐, ๐, ๐โช, where ๐, ๐, and ๐ are the vectorโs components. The displacement vector with initial point ๐1 = (๐ฅ1 , ๐ฆ1 , ๐ง1 ) and terminal point ๐2 = (๐ฅ2 , ๐ฆ2 , ๐ง2 ) is โโโโโโโโ ๐1 ๐2 = ฬ (๐ฅ2 โ ๐ฅ1 )๐ฬ + (๐ฆ2 โ ๐ฆ1 )๐ฬ + (๐ง2 โ ๐ง1 )๐. Example 2 โ Find the displacement vector โโโโโโโโ ๐1 ๐2 if ๐1 = (โ1,2,3) and ๐2 = (4,6,2). Let ๐ฃ = โฉ๐1 , ๐1 , ๐1 โช and ๐ค โโ = โฉ๐2 , ๐2 , ๐2 โช be two vectors and let ๐ผ be a scalar. Then: ๏ท ๏ท ๏ท ๏ท ๐ฃ=๐ค โโ if and only if ๐1 = ๐2 , ๐1 = ๐2 , and ๐1 = ๐2 . ๐ฃ±๐ค โโ = (๐1 ± ๐2 )๐ฬ + (๐1 ± ๐2 )๐ฬ + (๐1 + ๐2 )๐ฬ ๐ผ๐ฃ = (๐ผ๐1 )๐ฬ + (๐ผ๐1 )๐ฬ + (๐ผ๐1 )๐ฬ โ๐ฃ โ = โ๐12 + ๐12 + ๐12 Sullivan & Sullivan โ Section 9.6 10.3 Vectors in Space Example 3 โ If ๐ฃ = โฉ2,3, โ2โช and ๐ค โโ = โฉ3, โ4,5โช, find (a) ๐ฃ + ๐ค โโ and (b) ๐ฃ โ ๐ค โโ . Example 4 โ If ๐ฃ = โฉ2,3, โ2โช and ๐ค โโ = โฉ3, โ4,5โช, find (a) 3๐ฃ, (b) 2๐ฃ โ 3๐ค โโ , and (c) โ๐ฃโ. โ ๐ฃ Recall ๐ฃฬ = โ๐ฃโโ is a unit vector in the direction of ๐ฃ, and as a consequence ๐ฃ = โ๐ฃโ๐ฃฬ. Example 5 โ Find the unit vector in the same direction as โฉ2, โ3, โ6โช. If ๐ฃ = โฉ๐1 , ๐1 , ๐1 โช and ๐ค โโ = โฉ๐2 , ๐2 , ๐2 โช with an angle ๐ between them, then ๐ฃ โ ๐ค โโ = ๐1 ๐2 + ๐1 ๐2 + ๐1 ๐2 = โ๐ฃโโ๐ค โโ โ cos ๐. Sullivan & Sullivan โ Section 9.6 10.3 Vectors in Space Example 6 โ If ๐ฃ = โฉ2, โ3,6โช and ๐ค โโ = โฉ5,3, โ1โช, find (a) ๐ฃ โ ๐ค โโ , (b) ๐ค โโ โ ๐ฃ, (c) ๐ฃ โ ๐ฃ, (d) ๐ค โโ โ ๐ค โโ , (e) โ๐ฃโ, and (f) โ๐ค โโ โ. If ๐ข โ , ๐ฃ, and ๐ค โโ are vectors, then: ๏ท ๏ท ๏ท ๏ท ๐ข โ โ๐ฃ =๐ฃโ๐ข โ ๐ข โ โ (๐ฃ + ๐ค โโ ) = ๐ข โ โ๐ฃ+๐ข โ โ๐ค โโ 2 ๐ฃ โ ๐ฃ = โ๐ฃ โ โ0 โ ๐ฃ = 0 Example 7 โ Find the angle ๐ between โฉ2, โ3,6โช and โฉ2,5, โ1โช when placed tail-to-tail. A nonzero vector ๐ฃ in space can be described by specifying its magnitude and its three direction angles, ๐ผ, ๐ฝ, and ๐พ. These direction angles are defined as: ๏ท ๏ท ๏ท ๐ผ = the angle between ๐ฃ and ๐ฬ, 0 โค ๐ผ โค ๐ ๐ฝ = the angle between ๐ฃ and ๐ฬ, 0 โค ๐ฝ โค ๐ ๐พ = the angle between ๐ฃ and ๐ฬ, 0 โค ๐พ โค ๐ ๐ ๐ If ๐ฃ = โฉ๐, ๐, ๐โช is a nonzero vector in space, the direction angles ๐ผ, ๐ฝ, and ๐พ obey cos ๐ผ = โ๐ฃโโ, cos ๐ฝ = โ๐ฃโโ, ๐ and cos ๐พ = โ๐ฃโโ. Example 8 โ Find the direction angles of โฉโ3,2, โ6โช. Sullivan & Sullivan โ Section 9.6 10.3 Vectors in Space If ๐ผ, ๐ฝ, and ๐พ are the direction angles of a nonzero vector in space, then cos2 ๐ผ + cos 2 ๐ฝ + cos 2 ๐พ = 1. ๐ ๐ Example 9 โ The vector ๐ฃ makes an angle of ๐ผ = 3 with the positive ๐ฅ-axis, an angle of ๐ฝ = 3 with the positive ๐ฆ-axis, and an acute angle ๐พ with the positive ๐ง-axis. Find ๐พ. It can be shown that ๐ฃ = โ๐ฃ โ[(cos ๐ผ)๐ฬ + (cos ๐ฝ)๐ฬ + (cos ๐พ)๐ฬ]. 2 Example 10 โ Find the point that is 3 of the way from (5, โ2, โ6) to (11, โ11, 6). Example 11 โ Find the vector projection of โฉ3, 5, โ4โช onto โฉ1, โ7, 2โช. Sullivan & Sullivan โ Section 9.6 10.3 Vectors in Space Complete the following exercises on a separate sheet of paper. In exercises 1 โ 4, find the displacement vector โโโโโโ ๐ท๐ธ. 1. ๐ = (0,0,0) and ๐ = (โ3, โ5,4) 3. ๐ = (โ3,2,0) and ๐ = (6,5, โ1) 2. ๐ = (3,2, โ1) and ๐ = (5,6,0) 4. ๐ = (โ1,4, โ2) and ๐ = (6,2,2) โ โ. In exercises 5 โ 8, find โ๐ 5. ๐ฃ = โฉ3, โ6, โ2โช 7. ๐ฃ = โฉโ1, โ1,1โช 6. ๐ฃ = โฉโ6,12,4โช 8. ๐ฃ = โฉ6,2, โ2โช ฬ and ๐ ฬ. โ = ๐๐ฬ โ ๐๐ฬ + ๐๐ In exercises 9 โ 14, find each requested quantity if ๐ โโโ = โ๐๐ฬ + ๐๐ฬ โ ๐๐ 9. 2๐ฃ + 3๐ค โโ 12. โ๐ฃ + ๐ค โโ โ 10. 3๐ฃ โ 2๐ค โโ 13. โ๐ฃ โ โ โ๐ค โโ โ 11. โ๐ฃ โ ๐ค โโ โ 14. โ๐ฃ โ + โ๐ค โโ โ ฬ in the same direction as ๐ โ. In exercises 15 โ 20, find the unit vector ๐ 15. ๐ฃ = 5๐ฬ 18. ๐ฃ = โ6๐ฬ + 12๐ฬ + 4๐ฬ 16. ๐ฃ = โ2๐ฬ 19. ๐ฃ = ๐ฬ + ๐ฬ + ๐ฬ 17. ๐ฃ = 3๐ฬ โ 6๐ฬ โ 2๐ฬ 20. ๐ฃ = 2๐ฬ โ ๐ฬ + ๐ฬ โ โ๐ โ and ๐ In exercises 21 โ 24, find the dot product ๐ โโโ and the angle between ๐ โโโ . 21. ๐ฃ = ๐ฬ + ๐ฬ, ๐ค โโ = โ๐ฬ + ๐ฬ โ ๐ฬ 23. ๐ฃ = ๐ฬ + 3๐ฬ + 2๐ฬ , ๐ค โโ = ๐ฬ โ ๐ฬ + ๐ฬ 22. ๐ฃ = 2๐ฬ + 2๐ฬ โ ๐ฬ , ๐ค โโ = ๐ฬ + 2๐ฬ + 3๐ฬ 24. ๐ฃ = 3๐ฬ โ 4๐ฬ + ๐ฬ , ๐ค โโ = 6๐ฬ โ 8๐ฬ + 2๐ฬ โ = In exercises 25 โ 28, find the direction angles of each vector. Write each vector in the form ๐ ฬ โ๐ โ โ[(๐๐๐ ๐ถ)๐ฬ + (๐๐๐ ๐ท)๐ฬ + (๐๐๐ ๐ธ)๐]. 25. ๐ฃ = โฉโ6,12,4โช 27. ๐ฃ = โฉ0,1,1โช 26. ๐ฃ = โฉ1, โ1, โ1โช 28. ๐ฃ = โฉ2,3, โ4โช In exercises 29 โ 32, find the radius and center of each sphere. 29. ๐ฅ 2 + ๐ฆ 2 + ๐ง 2 + 2๐ฅ โ 2๐ง = 2 31. ๐ฅ 2 + ๐ฆ 2 + ๐ง 2 โ 4๐ฅ = 0 30. ๐ฅ 2 + ๐ฆ 2 + ๐ง 2 โ 4๐ฅ + 4๐ฆ + 2๐ง = 0 32. 3๐ฅ 2 + 3๐ฆ 2 + 3๐ง 2 + 6๐ฅ โ 6๐ฆ = 3 33. Find the work done by a force of 3 newtons acting in the direction 2๐ฬ + ๐ฬ + 2๐ฬ in moving an object 2 meters from (0,0,0) to (0,2,0). 34. Find the work done in moving an object along a vector ๐ข โ = 3๐ฬ + 2๐ฬ โ 5๐ฬ if the applied force is ๐นฬ = 2๐ฬ โ ๐ฬ โ ๐ฬ. Use meters for distance and newtons for force. 35. Find the point that is 75% of the way from (โ3,5,2) to (1, โ3,14). Sullivan & Sullivan โ Section 9.6 10.3 Vectors in Space 1 36. Find the point that is 3 of the way from (5,3, โ7) to (11,0, โ16). 37. Find the vector projection of โฉ5, โ2,1โช onto โฉ8,2, โ5โช. 38. Find the vector projection of โฉ7,2, โ6โช onto โฉโ3,5, โ1โช. Sullivan & Sullivan โ Section 9.6 10.4 The Cross Product and Planar Equations If ๐ฃ = โฉ๐1 , ๐1 , ๐1 โช and ๐ค โโ = โฉ๐2 , ๐2 , ๐2 โช are two vectors in space, then the cross product (or vector product) ๐ฃ×๐ค โโ = (๐1 ๐2 โ ๐2 ๐1 )๐ฬ โ (๐1 ๐2 โ ๐2 ๐1 )๐ฬ + (๐1 ๐2 โ ๐2 ๐1 )๐ฬ. Example 1 โ If ๐ฃ = โฉ2,3,5โช and ๐ค โโ = โฉ1,2,3โช, find ๐ฃ × ๐ค โโ . ๐ด 2 3 Example 2 โ Calculate | | and | 2 1 2 1 ๐ต 3 2 ๐ถ 5 |. 3 ๐ฬ The cross product of the vectors ๐ฃ = โฉ๐1 , ๐1 , ๐1 โช and ๐ค โโ = โฉ๐2 , ๐2 , ๐2 โช is ๐ฃ × ๐ค โโ = |๐1 ๐2 ๐ฬ ๐1 ๐2 ๐ฬ ๐1 |. ๐2 Example 3 โ If ๐ฃ = โฉ2,3,5โช and ๐ค โโ = โฉ1,2,3โช, find (a) ๐ฃ × ๐ค โโ , (b) ๐ค โโ × ๐ฃ, (c) ๐ฃ × ๐ฃ , and (d) ๐ค โโ × ๐ค โโ . Sullivan & Sullivan โ Section 9.7 Foerster โ Sections 10.5 & 10.6 10.4 The Cross Product and Planar Equations If ๐ข โ , ๐ฃ, and ๐ค โโ are vectors in space and if ๐ผ is a scalar, then ๏ท ๏ท ๏ท ๏ท ๏ท ๏ท ๏ท ๏ท ๐ข โ ×๐ข โ =0 ๐ข โ × ๐ฃ = โ(๐ฃ × ๐ข โ) ๐ผ(๐ข โ × ๐ฃ ) = (๐ผ๐ข โ)×๐ฃ =๐ข โ × (๐ผ๐ฃ ) ๐ข โ (๐ฃ + ๐ค โโ ) = (๐ข โ × ๐ฃ ) + (๐ข โ × ๐ฃ) ๐ข โ × ๐ฃ is orthogonal to both ๐ข โ and ๐ฃ (and normal to the plane that contains ๐ข โ and ๐ฃ ) โ๐ข โ × ๐ฃ โ = โ๐ข โ โโ๐ฃ โ sin ๐ where ๐ is the angle between ๐ข โ and ๐ฃ when placed tail-to-tail. โ and ๐ฃ โ 0 โ as adjacent sides. โ๐ข โ × ๐ฃ โ is the area of the parallelogram having ๐ข โ โ 0 1 โ and ๐ฃ โ 0 โ as adjacent sides. โ๐ข โ × ๐ฃ โ is the area of the triangle having ๐ข โ โ 0 2 ๏ท ๐ข โ × ๐ฃ = โ0 if and only if ๐ข โ and ๐ฃ are parallel. Example 4 โ Find a vector that is orthogonal to ๐ข โ = โฉ3, โ2,1โช and ๐ฃ = โฉโ1,3, โ1โช. Example 5 โ Find the area of the parallelogram whose vertices are ๐1 = (0,0,0), ๐2 = (3, โ2,1), ๐3 = (โ1,3, โ1), and ๐4 = (2,1,0). The general form of the equation of a plane in space is ๐ด๐ฅ + ๐ต๐ฆ + ๐ถ๐ง = ๐ท, where ๐โ = โฉ๐ด, ๐ต, ๐ถโช is a normal vector to the plane. Example 6 โ Find the equation of the plane containing the point (3,5,7) with normal vector ๐โ = โฉ11,2,13โช. Sullivan & Sullivan โ Section 9.7 Foerster โ Sections 10.5 & 10.6 10.4 The Cross Product and Planar Equations Example 7 โ Find two vectors โโโโ ๐1 and โโโโ ๐2 normal to the plane 7๐ฅ โ 3๐ฆ + 8๐ง = โ51. Example 8 โ Find an equation of the plane perpendicular to the segment connecting points ๐1 = (3,8, โ2) and ๐2 = (7, โ1,6) and passing through the point 30% of the way from point ๐1 to point ๐2 . Example 9 โ Find a particular equation of the plane containing the points ๐1 = (โ5,5,5), ๐2 = (โ3,2,7), and ๐3 = (1,12,6). Complete the following exercises on a separate sheet of paper. In exercises 1 โ 4, find โ๐ × ๐ โโโ and ๐ โโโ × โ๐. 1. ๐ฃ = โฉโ1,3,2โช, ๐ค โโ = โฉ3, โ2, โ1โช 3. ๐ฃ = โฉ3,1,3โช, ๐ค โโ = โฉ1,0, โ1โช 2. ๐ฃ = โฉ1, โ4,2โช, ๐ค โโ = โฉ3,2,1โช 4. ๐ฃ = โฉ2, โ3,0โช, ๐ค โโ = โฉ0,3, โ2โช Sullivan & Sullivan โ Section 9.7 Foerster โ Sections 10.5 & 10.6 10.4 The Cross Product and Planar Equations โ = โฉ๐, โ๐, ๐โช, ๐ โ = โฉโ๐, ๐, ๐โช, and ๐ In exercises 5 โ 14, find each expression given that ๐ โโโ = โฉ๐, ๐, ๐โช. 5. ๐ฃ × ๐ค โโ 10. (๐ข โ × ๐ฃ) โ ๐ค โโ 6. ๐ค โโ × ๐ข โ 11. ๐ฃ โ (๐ข โ ×๐ค โโ ) 7. ๐ฃ × 4๐ค โโ 12. (๐ข โ × ๐ฃ) × ๐ค โโ 8. โ3๐ข โ ×๐ค โโ 13. Find a vector orthogonal to both ๐ฃ and ๐ฬ + ๐ฬ. 9. ๐ฃ โ (๐ฃ × ๐ค โโ ) 14. Find a vector orthogonal to both ๐ข โ and ๐ฬ + ๐ฬ. In exercises 15 โ 16, find the area of the parallelogram with vertices ๐ท๐ , ๐ท๐ , ๐ท๐ , and ๐ท๐ . 15. ๐1 = (2,1,1), ๐2 = (2,3,1), ๐3 = (โ2,4,1), ๐4 = (โ2,6,1) 16. ๐1 = (โ1,1,1), ๐2 = (โ1,2,2), ๐3 = (โ3,4, โ5), ๐4 = (โ3,5, โ4) In exercises 17 โ 18, find the area of the triangle with vertices ๐ท๐ , ๐ท๐ , and ๐ท๐ . 17. ๐1 = (0,0,0), ๐2 = (2,3,1), ๐3 = (โ2,4,1) 18. ๐1 = (โ2,0,2), ๐2 = (2,1, โ1), ๐3 = (2, โ1,2) In exercises 19 โ 20, find two normal vectors to the plane, pointing in opposite directions. 19. 3๐ฅ + 5๐ฆ โ 7๐ง = โ13 20. 4๐ฅ โ 7๐ฆ + 2๐ง = 9 In exercises 21 -26, find a particular equation of the plane described. 21. Perpendicular to ๐โ = โฉ3, โ5,4โช, containing the point (6, โ7, โ2). 22. Perpendicular to ๐โ = โฉโ1,3, โ2โช, containing the point (4,7,5). 23. Perpendicular to the line segment connecting the points (3,8,5) and (11,2, โ3) and passing through the midpoint of the segment. 24. Parallel to the plane 3๐ฅ โ 7๐ฆ + 2๐ง = 11 and containing the point (8,11, โ3). 25. Parallel to the plane 5๐ฅ โ 3๐ฆ โ ๐ง = โ4 and containing the point (4, โ6,1). 26. Perpendicular to ๐โ = โฉ4,3, โ2โช and having ๐ฅ-intercept 5. In exercises 27 โ 29, find a particular equation of the plane containing the given points. 27. (3,5,8), (โ2,4,1), (โ4,7,3) 29. (0,3, โ7), (5,0, โ1), (4,3,9) 28. (5,7,3), (4, โ2,6), (2, โ6,1) 30. The cross product of the normal vectors to two planes is a vector that points in the direction of the line of intersection of the planes. Find a particular equation of the plane containing the point (โ3,6,5) and normal to the line of intersection of the planes 3๐ฅ + 5๐ฆ + 4๐ง = โ13 and 6๐ฅ โ 2๐ฆ + 7๐ง = 8. Sullivan & Sullivan โ Section 9.7 Foerster โ Sections 10.5 & 10.6 10.5 Plane Curves and Parametric Equations Let ๐ and ๐ be continuous functions of ๐ก on an interval ๐ผ. The set of all points (๐ฅ, ๐ฆ), where ๐ฅ = ๐(๐ก) and ๐ฆ = ๐(๐ก) is called a plane curve. The variable ๐ก is called a parameter, and the equations that define ๐ฅ and ๐ฆ are called parametric equations. A pair of parametric equations that describe a given curve is called a parameterization of the curve. More than one parameterization is possible for a given curve. Example 1 โ Find three parameterizations of the line through (1, โ3) with slope โ2. Example 2 โ By hand, graph the curve given by ๐ฅ = โ2๐ก and ๐ฆ = 4๐ก 2 โ 4, โ1 โค ๐ก โค 2. Example 3 โ Consider the curve given by ๐ฅ = โ2๐ก and ๐ฆ = 4๐ก 2 โ 4, โ1 โค ๐ก โค 2. Find an equation in ๐ฅ and ๐ฆ whose graph includes the graph of the given curve. Example 4 โ Eliminate the parameter in the equations ๐ฅ = 5 cos ๐ก + 4, ๐ฆ = 2 sin ๐ก โ 3, 0 โค ๐ก โค 2๐. Holt โ Sections 11.7 & 11.7a 10.5 Plane Curves and Parametric Equations Example 5 โ Convert (๐ฆโ5)2 9 โ (๐ฅ+2)2 16 = 1 into parametric equations. Example 6 โ Given the parent relation ๐ฅ = ๐ฆ 2, write a set of parametric equations to represent the relation, and sketch the graph. Then write parametric equations of the following successive transformations of the parent relation, and sketch each graph: (a) a horizontal dilation by a factor of 5; (b) then a horizontal shift 3 units to the right; (c) then a vertical shift down 2 units. When a projectile ๏ท ๏ท ๏ท ๏ท Is fired from the position (0, ๐) on the positive ๐ฆ-axis at an angle of ๐ with the horizontal, In the direction of the positive ๐ฅ-axis, With initial velocity ๐ฃ feet per second, With negligible air resistance, Then its position at time ๐ก seconds is given by the parametric equations ๐ฅ = (๐ฃ cos ๐)๐ก and ๐ฆ = (๐ฃ sin ๐)๐ก + ๐ โ 16๐ก 2 . If the initial velocity is given as ๐ฃ meters per second, the ๐ฆ equation becomes ๐ฆ = (๐ฃ sin ๐)๐ก + ๐ โ 4.9๐ก 2 . Example 5 โ A golfer hits a ball with an initial velocity of 140 feet per second so that its path as it leaves the ground makes an angle of 31° with the horizontal. (a) When does the ball hit the ground? (b) How far from its starting point does it land? (c) What is the maximum height of the ball during its flight? Holt โ Sections 11.7 & 11.7a 10.5 Plane Curves and Parametric Equations Example 6 โ A batter hits a ball that is 3 feet above the ground. The ball leaves the bat with an initial velocity of 138 feet per second, making an angle of 26° with the horizontal and heading toward a 25-foot fence that is 400 feet away. Will the ball go over the fence? Complete the following exercises on a separate sheet of paper. In exercises 1 โ 6, the given curve is part of the graph of an equation in ๐ and ๐. Eliminate the parameter. 1. ๐ฅ = ๐ก + 5, ๐ฆ = 2๐ก + 1, ๐ก โฅ 0 4. ๐ฅ = โ2 + ๐ก 2 , ๐ฆ = 1 + 2๐ก 2 , for any ๐ก 2. ๐ฅ = ๐ก + 5, ๐ฆ = โ๐ก, ๐ก โฅ 0 5. ๐ฅ = 4 sin 2๐ก , ๐ฆ = 2 cos 2๐ก , 0 โค ๐ก โค 2๐ 3. ๐ฅ = ๐ก 2 + 1, ๐ฆ = ๐ก 2 โ 1, for any ๐ก 6. ๐ฅ = 2 sin ๐ก โ 3, ๐ฆ = 2 cos ๐ก + 1, 0 โค ๐ก โค 2๐ In exercises 7 โ 16, find parametric equations for the curve whose equation is given. 7. ๐ฆ2 49 + ๐ฅ2 81 12. =1 8. ๐ฅ 2 + 4๐ฆ 2 = 1 9. ๐ฆ2 9 16 + (๐ฆ+3)2 12 + (๐ฆ+2)2 12 =1 14. ๐ฅ = โ3(๐ฆ โ 1)2 โ 2 ๐ฅ2 โ 16 = 1 (๐ฅโ2)2 4 13. ๐ฆ = 3(๐ฅ โ 2)2 โ 3 15. 10. 2๐ฅ 2 โ ๐ฆ 2 = 4 11. (๐ฅ+5)2 =1 16. (๐ฆ+1)2 9 (๐ฆ+5)2 9 โ โ (๐ฅโ1)2 25 (๐ฅโ2)2 1 =1 =1 In exercises 17 โ 18, sketch the graphs of the given curves and compare them. Do they differ? If so, how? 17. 18. a. ๐ฅ = โ4 + 6๐ก, ๐ฆ = 7 โ 12๐ก, 0 โค ๐ก โค 1 a. ๐ฅ = ๐ก, ๐ฆ = ๐ก 2 , for any ๐ก b. ๐ฅ = 2 โ 6๐ก, ๐ฆ = โ5 + 12๐ก, 0 โค ๐ก โค 1 b. ๐ฅ = โ๐ก, ๐ฆ = ๐ก, for any ๐ก c. ๐ฅ = ๐ ๐ก , ๐ฆ = ๐ 2๐ก , for any ๐ก In exercises 19 โ 20, find a parameterization of the given curve. 19. Line segment from (14, โ5) to (5, โ14) Holt โ Sections 11.7 & 11.7a 20. Line segment from (18,4) to (โ16,14) 10.5 Plane Curves and Parametric Equations In exercises 21 โ 26, assume that air resistance is negligible. 21. A ball is thrown from a height of 5 feet above the ground with an initial velocity of 60 feet per second at an angle of 50° with the horizontal. When and where does the ball hit the ground? 22. A medieval bowman shoots an arrow which leaves the bow 4 feet above the ground with an initial velocity of 88 feet per second with an angle of elevation of 48°. Will the arrow go over the 40-foot castle wall that is 200 feet from the archer? 23. A golfer at a driving range stands on a platform 2 feet above the ground and hits the ball with an initial velocity of 120 feet per second at an angle of elevation of 39°. There is a 32-foot-high fence 400 feet away. Will the ball fall short, hit the fence, or go over it? 24. A golf ball is hit off the tee at an angle of 30° and lands 300 feet away. What was its initial velocity? 25. A football kicked from the ground has an initial velocity of 75 feet per second. a. Find the angle needed for the ball to travel exactly 150 feet. b. Set up the parametric equations that describe the ballโs path. 26. A skeet is fired from the ground with an initial velocity of 110 feet per second at an angle of 28°. a. How long is the skeet in the air? b. How high does it go? Holt โ Sections 11.7 & 11.7a Module 10 โ Selected Solutions