Trig/Math Anal



Trig/Math Anal Name_______________________No_____

|LATE AND ABSENT HOMEWORK IS ACCEPTED UP TO THE TIME OF THE CHAPTER TEST ON _____ |

|HW NO. |SECTIONS |ASSIGNMENT |DUE |√ |

| |(Brown Book) | | | |

|V-1 |12-1/12-2 |Practice Set A | | |

|V-2 |12-3 |Practice Set B #1-12 | | |

|V-2B |12-3 |Practice Set G | | |

|V-3 |12-4 |Practice Set C | | |

|V-3B |12-4 |Practice Set K | | |

|V-4 |12-5 |Practice Set D | | |

|V-5 |12-6 |Practice Set E | | |

|V-6 |12-9 |Practice Set F #1-3 | | |

|V-7 |Review |Practice Set F #4-17 | | |

|Practice Set A: Vectors |

|1. UNavigation: U A ship travels 200 km west from port and then 240 km due south before it is disabled. Illustrate this in a vector diagram. |

|Use trigonometry to find the course that a rescue ship must take from port in order to reach the disabled ship. |

|2. UAviation:U On graph paper, make a diagram that illustrates the velocity of an airplane heading east at 400 knots. Illustrate a wind |

|velocity of 50 knots blowing toward the northeast. If the airplane encounters this wind, illustrate its resultant velocity. Estimate the |

|resultant speed and direction of the airplane. (The direction is the angle the resultant makes with due north measured clockwise from north.) |

|Plot points A and B. Give the component form of [pic] and find [pic] |

|3. [pic] |4. [pic] |

|Polar coordinates of point P are given and O is the origin. Draw vector [pic] and give its component form. |

|5. [pic] |6. [pic] |

|7. Let [pic]. Calculate each expression. |

|a. [pic] b. [pic] c. [pic] d. [pic] |

|Find the coordinates of the point P described. |

|8. [pic] of the way from A to B. |

|9. [pic] of the way from A to B. |

|10. UPhysics:U Suppose that you pull a child in a wagon by pulling a rope that makes a[pic] angle with the ground. |

|a. If the pulling force F is 40 lb in the direction of the rope, give the horizontal and vertical components of the force. |

|b. Which component, horizontal or vertical, moves the wagon along the ground? |

|c. If the rope made a[pic] angle with the ground rather than a [pic] angle, would the wagon move more easily or less easily? Why? |

|Practice Set B: Vectors and Parametric Equations |

|Find vector and parametric equations for each specified line. |

|1. The line through (1, 5) with direction vector (2, -1) |2. The line through (1, 0) and (3, -4) |

|3. The line through (-2, 3) and (5, 1) |4. The horizontal line through [pic] |

|5. A point moves in the plane so that its position [pic] at time t is given by the equation [pic] |

|Graph the point’s position at the times [pic] |

|Find the velocity and speed of the moving point. |

|Find the parametric equations of the moving point. |

|6. Find the vector and parametric equations of the moving object with velocity (3, -1) and position vector at time [pic] is (2, 3). |

|7. A line has vector equation[pic]. Give a pair of parametric equations and a Cartesian equation of the line. |

|8. A line has parametric equations [pic] and[pic]. Give a vector equation and a Cartesian equation of the line. |

|9. a. Describe the line having parametric equations [pic] and[pic]. |

|Give a direction vector of the line. |

|What can you say about the slope of the line? |

|10. a. A line has direction vector (2, 3). What is the slope of the line? |

|A line has direction vector (4, 6). What is the slope of the line? |

|Explain why the following lines are parallel: |

|Find a vector equation of the line through (7, 9) and parallel to these lines. |

|11. At time t, the position of an object moving with constant velocity is given by the parametric equations [pic] and[pic]. |

|a. What are the velocity and speed of the object? |

|b. When and where does it cross the line [pic]. |

|12. An object moves with constant velocity so that its position at time t is[pic]. When and where does the object cross the circle[pic]? |

|Illustrate with a sketch. |

|13. Without graphing, describe the curve with parametric equations [pic] and[pic]. (Hint: What is the value of[pic]?) |

|14. a. Before graphing, describe what you think the curve with parametric equations [pic] and [pic] looks like. Then graph the |

|equations. |

|Find a Cartesian equation for the parametric equations given in part (a) |

|An ellipse has Cartesian equation[pic]. What do you think the parametric equations of the ellipse are? Check your answer by graphing. |

|Practice Set C: Parallel and Perpendicular Vectors; Dot Products |

|1. Find: a. [pic] b. [pic] |

|2. Find the value of a if the vectors (4, 6) and (a, 3) are |

|a. parallel b. perpendicular |

|3. If [pic], find: a. [pic] b. [pic] |

|4. If [pic], verify that |

|a. [pic] b. [pic] |

|5. Verify that [pic] if [pic] |

|6. Verify that the angle between [pic] and [pic] is[pic]. |

|7. Find the measure of the angle between [pic] and[pic]. |

|8. Given A(1, 5), B(4, 6), and C(2, 8), find the measure of [pic]. |

|9. Given P(0, 3), Q(2, 4), and R(3, 7), verify that [pic]. |

|10. a. Given A(1, -3), B(-1, 3) and C(6, 2), find [pic]. |

|b. Use the formula [pic] to find the area of [pic] |

|Practice Set D: Vectors in Three Dimensions |

|1. Find the length and midpoint of [pic]: A=(2, 5, -3) and B=(0, 3, 1) |

|2. Simplify |

|a. [pic] b. [pic] c. [pic] |

|3. Find the value of [pic]if [pic] and [pic] are perpendicular. |

|4. Find an equation of the sphere with radius 7 and center (1, 5, 3). Show that point [pic] is on the sphere. |

|5. Find the center and radius of the sphere with equation [pic] |

|6. Find the angle between (8, 6, 0) and (2, -1, 2) to the nearest tenth of a degree. |

|7. Let[pic], [pic], and [pic]. |

|a. Show that [pic] and [pic] are perpendicular. b. Find the area of right triangle ABC |

|8. Line L has vector equation [pic] |

|a. Find three parametric equations of L. |

|b. Name two points on L. |

|c. Write a vector equation of the line containing (1, 2, 3) and parallel to L |

|9. Write vector and parametric equations for the line containing [pic] and[pic]. |

|10. Where does the line [pic] intersect |

|a. the xy-plane b. the yz-plane c. the xz-plane |

|11. Describe the set of points S in the xy-plane that are also on the sphere whose equation is[pic]. Give an equation of S. |

|12. Show that the lines with equations [pic] and [pic] intersect. Find the coordinates of their point of intersection. |

|Practice Set E: Vectors and Planes |

|1. Sketch the plane: |

|a. [pic] b. [pic] |

|2. Find a vector perpendicular to the plane whose equation is [pic] |

|3. Find a Cartesian equation of the plane: |

|Vector (2, 3, 5) is perpendicular to the plane that contains point A(3, 1, 7) |

|Vector (1, -4, 2) is perpendicular to the plane that contains point A(3, 0, 2) |

|4. Consider the points A(2, 2, 2) and B(4, 6, 8) |

|a. Find a Cartesian equation of the plane that is perpendicular to [pic] at its midpoint M. |

|b. Show that the point P(2,0,8) satisfies your answer to part (a) |

|5. Find an equation of the plane tangent to the sphere [pic] at the point (7, -1, 4). |

|6. The plane [pic] intersects the sphere [pic] in a circle. Find the area of the circle. |

|7. To the nearest tenth of a degree, find the measure of the angle between the planes [pic] and [pic]. |

|8. Are the planes [pic] and [pic] perpendicular? |

|9. Which of the following planes are perpendicular and which are parallel? |

|[pic] [pic] [pic] |

|Practice Set F: Determinants and Vectors |

|1. Let [pic] |

|a. Calculate [pic]. Do your results agree with property 2? |

|b. Verify that [pic]is perpendicular to [pic]. |

|c. Find the area of the parallelogram determined by [pic]. |

|2. Let [pic] |

|a. Find a vector perpendicular to the plane determined by [pic]. |

|b. Find a Cartesian equation of the plane determined by [pic]. |

|3. Angle [pic] is between vectors [pic] |

|a. Find [pic] by using property 3 of the cross product |

|b. Find [pic]by using the dot product property |

|c. Verify that [pic] |

|Review Section: |

|4. An object is pulled due south by a force [pic] of 5 N, and due east by a force [pic] of 12 N. Find the direction and magnitude of [pic] |

|5. Find the component form of [pic] and find [pic]if [pic] |

|6. Given polar coordinates [pic] draw [pic] and find its component form. |

|7. Find the coordinates of the point P, [pic] of the way from [pic]to [pic] |

|8a. Find the velocity and speed of an object that moves with constant velocity so that its position at time [pic] is [pic] |

|8b. Find a pair of parametric equations of the path of the object. |

|8c. When and where does it intersect the parabola[pic]? |

|9. Find vector and parametric equations for the line through [pic] |

|10. Find a Cartesian equation for the vector equation [pic] |

|11. Find the value of [pic]if vectors [pic]are |

|a. parallel b. perpendicular |

|12. If [pic] |

|a. find [pic] b. find the angle between [pic] |

|13. Given[pic], find [pic]. |

|14. Given the points [pic] |

|a. find the length of [pic] b. find the midpoint of [pic] |

|15. Line L has equation [pic] |

|a. Write a vector equation for the line through (2,1,7) parallel to L. |

|b. Where does L intersect the xz-plane? |

|16. Find an equation of the plane that is tangent to the sphere with equation [pic] at the point (2,-3,-2). |

|17. [pic]determine a plane. |

|a. find a vector perpendicular to the plane b. find the area of triangle ABC |

|Practice Set G: Vectors and Parametric Equations |

|Find [pic] and [pic]: |

|1. [pic] |2. [pic] |

|Draw vector [pic] and give its component form: |

|3. [pic] |4. [pic] |

|5. Let [pic]. Calculate each expression. |

|a. [pic] b. [pic] c. [pic] d. [pic] |

|6. Find the coordinates of the point P [pic] of the way from A(-2, 4) to B(7, -2) |

|7. Find the coordinates of the point P [pic] of the way from A(-6, 5) to B(2, 9) |

|Find vector and parametric equations for each specified line. |

|8. The line through (-1, 4) and (5, 8) |9. The vertical line through (2, -3) |

|10. A point moves in the plane so that its position [pic] at time t is given by the equation [pic] |

|Graph |

|Find the velocity and speed of the moving point. |

|Find the parametric equations of the moving point. |

|11. A line has vector equation [pic]. Give a pair of parametric equations and a Cartesian equation of the line. |

|12. A line has parametric equations [pic]. Write a vector and Cartesian equation for the line. |

|13. Line L has equation [pic]. |

|a. Write a vector equation for the line through (2,1) parallel to L. |

|b. Write a vector equation for the line through (4, -3) perpendicular to L. |

|14. The velocity of a plane heading west is 525 knots. It encounters a wind heading north east with a velocity of 25 knots. Calculate the |

|resultant speed and direction of the plane. |

|Practice Set K: Vectors and Parametric Equations |

|1. Find [pic] and [pic]: [pic] |2. Draw vector [pic] and give its component form: [pic] |

|3. Let [pic]. Calculate each expression. |

|a. [pic] b. [pic] c. [pic] d. [pic] e. [pic] |

|4. Find the coordinates of the point P [pic] of the way from A(-4, 3) to B(2, -15) |

|Find vector and parametric equations for each specified line. |

|5. The line through (-5, 7) and (3, -2) |6. The horizontal line through (5, -3) |

|7. A point moves in the plane so that its position [pic] at time t is given by the equation [pic] |

|Graph |

|b. Find the velocity and speed of the moving point. |

|8. A line has vector equation [pic]. Give a pair of parametric equations and a Cartesian equation of the line. |

|9. An object moves with constant velocity so that its position at time t is [pic]. When and where does the object cross the line[pic]? |

|10. Find the value of a if the vectors (3a, 4) and (5, 7) are |

|a. parallel b. perpendicular |

|11. If [pic] find [pic] |

|12. Find the measure of the angle between [pic] and [pic]. |

|13. Given A(2, 7), B(-3, 6), and C(5, -1), find the measure of [pic]. |

|14. Given P(5, -2), Q(-3, 4), and R(6, 1), find the measure of [pic]. |

|15. Line L has equation [pic]. |

|a. Write a vector equation for the line through (3,-5) parallel to L. |

|b. Write a vector equation for the line through (2, -3) perpendicular to L. |

|16. The velocity of a plane heading south is 340 knots. It encounters a wind heading south west with a velocity of 65 knots. Calculate the |

|resultant speed and direction of the plane. |

|ANSWERS |

|Practice Set A |

|1. [pic] |2. 436.8 knots; [pic] |3. (2, 0); 2 |4. (-2, 6); [pic] |

|5. (1.85, 5.71) |6. (-1, -1.73) |7a. (-5, 5) |7b. (11, -3) |

|7c. (3, 1) |7d. [pic] |8. [pic] |9. (3, 6) |

|10a. [pic] |10b. horizontal |10c. easier | |

|Practice Set B |

|1. [pic] |2. [pic] |

|3. [pic] |4. [pic] |

|5b. [pic] |5c. [pic] |

|6. [pic] |7. [pic] |

|8. [pic] |9a. vertical line thru (2, 0) |

|9b. (0, 1) |9c. slope is undefined |10a. 1.5 |10b. 1.5 |

|10c. slopes are equal |10d. [pic] |

|11a. [pic] |11b. [pic] |

|12. [pic] |13. circle; radius=r; center (0,0) |

|14a. ellipse; vertices [pic] |14b. [pic] |

|14c. [pic] | |

|Practice Set C |

|1a. -7 |1b. 1 |2. 2; -4.5 |3. 13; 13 |

|7. [pic] |8. [pic] |10a. [pic] |10b. 20 |

|Practice Set D |

|1. [pic]; (1, 4, -1) |2a. (11, 6, 2) |2b. -5 |2c. [pic] |

|3. k=5 |5. [pic] |6. [pic] |7b. 9 |

|8a. [pic]; [pic]; [pic] |8b. ex: [pic]; [pic] |

|8c. [pic] |

|9. Vector: [pic]; Parametric: [pic] |

|10a. (3,-3,0) |10b. (0,-1,-4) |10c. [pic] |11. [pic] |12. (-2, 3, 1) |

|Practice Set E |

|2. (3, 4, 6) |3a. [pic] |3b. [pic] |4a. [pic] |

|5. [pic] |6. [pic] |7. [pic] |9. [pic];[pic] |

|Practice Set F |

|1c. [pic] |2a. (-1,4,1) |2b. [pic] |3a. 0.745 |

|3b. 0.667 |4.[pic], 13N |5. [pic] |6. (-6.14,5.14) |

|7. (2,2) |8a. [pic] |8b. [pic] |8c. [pic] |

|9. [pic] |10. [pic] |11a. -16 |

|11b. 4 |12a. -5 |12b. [pic] |13. [pic] |

|14a. 15.75 |14b. (-1,-1,1) |15a. [pic] |

|15b. (-4,0,9) |16. [pic] |17a. (-16,46,-24) |17b. 27.1 |

|Practice Set G |

|1. [pic] |2. [pic]; [pic] |

|3. [pic] |4. [pic] |5a. [pic] |5b. [pic] |

|5c. [pic] |5d. 33.29 |6. [pic] |7. (0, 8) |

|8. [pic]; [pic] |

|9. [pic]; [pic] |

|10b. velocity = [pic]; speed = [pic] |10c. [pic] |

|11. [pic]; [pic] |12. [pic] |

|13a. [pic] |13b. [pic] |

|14. [pic]; [pic] | |

|Practice Set K |

|1. [pic] |2. [pic] |3a. [pic] |3b. [pic] |

|3c. [pic] |3d. [pic] |3e. -12 |4. [pic] |

|5. [pic]; [pic] |

|6. [pic]; [pic] |

|7b. velocity = [pic]; speed = [pic] |8. [pic]; [pic] |

|9. when [pic]; where [pic] |10a. [pic] |10b. [pic] |

|11. 144 |12. [pic] |13. [pic] |14. [pic] |

|15a. [pic] |15b. [pic] |

|16. [pic] | |

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