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Geometry Vectors

Geometry Vectors. Warm up. Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. 1) BC 2) m /B 3) m /C. Vectors. You can think of a vector as a directed line segment. The vector below line segment. The

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Geometry Vectors

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  1. Geometry Vectors CONFIDENTIAL

  2. Warm up Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. 1) BC 2) m/B 3) m/C CONFIDENTIAL

  3. Vectors You can think of a vector as a directed line segment. The vector below line segment. The vector below may be named AB or v CONFIDENTIAL

  4. D 3 C 2 A vector can also be named using component from. The Component from4{x, y} of a vector lists the horizontal and vertical change from the initial point to the terminal point. The Component from of CD is {2,3}. CONFIDENTIAL

  5. Writing Vectors in Component From Subtract the coordinates of the initial point from the coordinates of the terminal point. Substitute the coordinate of the given points simplify. CONFIDENTIAL

  6. Now you try! 1) Write each vector in component from. a) b) The vector with initial point L(-1,1) and terminal point M(6,2) CONFIDENTIAL

  7. The magnitude of a vector is its length. The magnitude is written |AB| or | v |. When a vector is used to represent speed in a given direction, the magnitude of the vector equals the speed. For example, if a vector represents the course a kayaker paddles, the magnitude of the vector is the kayaker’s speed. CONFIDENTIAL

  8. Finding the magnitude of a Vector Step 1 Draw the vector on a coordinate plane. Use the origin as the initial point. Then (4,-2) is the terminal point. Step 2 Find the magnitude. Use the Distance Formula. CONFIDENTIAL

  9. Now you try! 2)Draw the vector (-3,1) on a coordinate plane. Fine its magnitude to the nearest tenth. CONFIDENTIAL

  10. The direction of a vector is the angle that it makes with a horizontal line. This angle is measured counterclockwise from the positive x -axis. The direction of AB is 60˚. CONFIDENTIAL

  11. The direction of a vector can also be given as a bearing relative to the compass directions north, south, east,and west. AB has a bearing of N 30˚ E. N B 30˚ W E A S CONFIDENTIAL

  12. Finding the Direction of a vector A wind velocity is given by the vector (2,5). Draw the vector on a coordinate plane. Find the direction of the vector to the nearest degree. Step 1 Draw the vector on a coordinate plane. Use the origin as the initial point. Next Page -> CONFIDENTIAL

  13. Step 2 Find the direction. Draw right triangle ABC as shown. /A is the angle formed by the vector and the x –axis, and CONFIDENTIAL

  14. Now you try! 3) The force exerted by a tugboat is given by the vector (7,3). Draw the vector on a coordinate plane. Find the direction of the vector to the nearest degree. CONFIDENTIAL

  15. Two vectors are equal vector if they have the same magnitude and the same direction For example, u = v. Equal vector do not have to have the same initial point and terminal point. | u | = | v | = 2√5 v Next Page -> CONFIDENTIAL

  16. Two vectors are parallel vectors if they have the same direction or if they have opposite directions. They may have different magnitudes. For example, w || x . equal vectors are always parallel vectors. |w| = 2√5 |x| =√5 x w CONFIDENTIAL

  17. Identifying Equal and Parallel Vectors Identify each of the following. A) AB = GH Identify vectors with the same magnitude and direction. B) Parallel vectors AB || GH and CD || EF Identify vectors with the same or opposite directions. CONFIDENTIAL

  18. Now you try! • 4) Identify each of the following. • Equal vectors • Parallel vectors M R P N Y S X Q CONFIDENTIAL

  19. The resultant vector is the vector that represents the sum of two given vectors. To add two vectors geometrically, you can use the head-to-tail method or the parallelogram method. CONFIDENTIAL

  20. Vector Addition METHOD EXAMPLE Head-to-tail method Place the initial point (tail) of the second vector. On the terminal point (head) of the first vector. The resultant is the vector that joins the initial point of the first vector to the terminal point of the second vector. u = v v u Next Page -> CONFIDENTIAL

  21. Vector Addition METHOD EXAMPLE Parallelogram Method Use the same initial point for both of the given vectors. Create a parallelogram by adding a copy of each vector at the terminal point (head) of the other vector. The resultant vector is a diagonal of the parallelogram formed. u = v v u CONFIDENTIAL

  22. To add vectors numerically, and their components. If u = (x ,y ) and v = (x ,y ), then u + v =(x +x , y +y ). 1 1 2 2 1 2 1 2 CONFIDENTIAL

  23. Sports Application A kayaker leaves shore at a bearing of N 55˚ E and paddles at a constant speed of 3 mi/h. There is 1 mi/h current moving due east. What are the kayak’s actual speed and direction? Round the speed to the nearest tenth and the direction to the nearest degree. Step 1Sketch vectors for the kayaker and the current. N N kayaker Current 3 y 55˚ 35˚ W E W E 1 X S S CONFIDENTIAL Next Page ->

  24. Step 2Write the vector for the kayaker in component from. The kayaker’s vector has a magnitude of 3 mi/h and makes an angle of 35˚ with the x –axis. N N kayaker Current 3 y 55˚ 35˚ W E W E 1 X S S CONFIDENTIAL Next Page ->

  25. Step 3 Write the vector for the current in component from. Since the current moves 1 mi/h in the direction of the x –axis, it has a horizontal component of 1 and vertical component of 0. So its vector is (1,0). N N Current kayaker 3 y 55˚ 35˚ W E E W 1 X S S Next Page -> CONFIDENTIAL

  26. Step 4 Find and sketch the resultant vector AB. Add the components of the kayaker’s vector and the current’s vector. (2.5, 1.7) + (1,0) = (3.5, 1.7) The resultant vector in the component from is (3.5, 1.7). N resultant B (3.5, 1.7) 1.7 A W E C 3.5 S Next Page -> CONFIDENTIAL

  27. Step 5 Find the magnitude and direction of the resultant vector. The magnitude of the resultant vector is the kayak’s actual speed. The angle measure formed by the resultant vector gives the kayak’s actual direction. N resultant B (3.5, 1.7) 1.7 A W E C 3.5 S CONFIDENTIAL

  28. Now you try! 5)What if….? Suppose the kayaker in Example 5 instead Paddles at 4 mi/h at a bearing of N 20˚ E. What are the kayak’s actual speed and direction? Round the speed to the nearest tenth and the direction to the nearest degree. CONFIDENTIAL

  29. Now some practice problems for you! CONFIDENTIAL

  30. Assessment Write each vector in component from. 1) PQ 2) AC with A(1,2) and C(6,5) Q P CONFIDENTIAL

  31. Draw each vector on a coordinate plane. Find its magnitude to the nearest tenth. 3) (1,4) 4) (-3,-2) 6) (5, -3) CONFIDENTIAL

  32. Draw each vector on a coordinate plane. Find the direction of the vector to the nearest degree. 6) A river’s current is the given by the vector (4,6). 7) The path of a hiker is given by the vector(6,3). CONFIDENTIAL

  33. Identify each of the following. 8) Equal vectors in diagram 1 9) Parallel vectors in diagram 2 Diagram 1 Diagram 2 Q G X B N D R H F C A M Y E P S CONFIDENTIAL

  34. 10) To reach a campsite, a hiker first walks for 2 mi at a bearing of N 40˚ E. Then he walks 3 mi due east. What are the magnitude and direction of his hike from his starting point to the campsite? Round the distance to the nearest tenth of a mile and the direction to the nearest degree. N 3 mi campsite 2 mi 40˚ W E S CONFIDENTIAL

  35. Let’s review Vectors You can think of a vector as a directed line segment. The vector below line segment. The vector below may be named AB or v CONFIDENTIAL

  36. D 3 C 2 A vector can also be named using component from. The Component from36{x, y} of a vector lists the horizontal and vertical change from the initial point to the terminal point. The Component from of CD is {2,3}. CONFIDENTIAL

  37. Writing Vectors in Component From Subtract the coordinates of the initial point from the coordinates of the terminal point. Substitute the coordinate of the given points simplify. CONFIDENTIAL

  38. The magnitude of a vector is its length. The magnitude is written |AB| or | v |. When a vector is used to represent speed in a given direction, the magnitude of the vector equals the speed. For example, if a vector represents the course a kayaker paddles, the magnitude of the vector is the kayaker’s speed. CONFIDENTIAL

  39. Finding the magnitude of a Vector Step 1 Draw the vector on a coordinate plane. Use the origin as the initial point. Then (4,-2) is the terminal point. Step 2 Find the magnitude. Use the Distance Formula. CONFIDENTIAL

  40. The direction of a vector is the angle that it makes with a horizontal line. This angle is measured counterclockwise from the positive x -axis. The direction of AB is 60˚. CONFIDENTIAL

  41. The direction of a vector can also be given as a bearing relative to the compass directions north, south, east,and west. AB has a bearing of N 30˚ E. N B 30˚ W E A S CONFIDENTIAL

  42. Finding the Direction of a vector A wind velocity is given by the vector (2,5). Draw the vector on a coordinate plane. Find the direction of the vector to the nearest degree. Step 1 Draw the vector on a coordinate plane. Use the origin as the initial point. Next Page -> CONFIDENTIAL

  43. Step 2 Find the direction. Draw right triangle ABC as shown. /A is the angle formed by the vector and the x –axis, and CONFIDENTIAL

  44. Two vectors are equal vector if they have the same magnitude and the same direction For example, u = v. Equal vector do not have to have the same initial point and terminal point. | u | = | v | = 2√5 v Next Page -> CONFIDENTIAL

  45. Two vectors are parallel vectors if they have the same direction or if they have opposite directions. They may have different magnitudes. For example, w || x . equal vectors are always parallel vectors. |w| = 2√5 |x| =√5 x w CONFIDENTIAL

  46. Identifying Equal and Parallel Vectors Identify each of the following. A) AB = GH Identify vectors with the same magnitude and direction. B) Parallel vectors AB || GH and CD || EF Identify vectors with the same or opposite directions. CONFIDENTIAL

  47. The resultant vector is the vector that represents the sum of two given vectors. To add two vectors geometrically, you can use the head-to-tail method or the parallelogram method. CONFIDENTIAL

  48. Vector Addition METHOD EXAMPLE Head-to-tail method Place the initial point (tail) of the second vector. On the terminal point (head) of the first vector. The resultant is the vector that joins the initial point of the first vector to the terminal point of the second vector. u = v v u Next Page -> CONFIDENTIAL

  49. Vector Addition METHOD EXAMPLE Parallelogram Method Use the same initial point for both of the given vectors. Create a parallelogram by adding a copy of each vector at the terminal point (head) of the other vector. The resultant vector is a diagonal of the parallelogram formed. u = v v u CONFIDENTIAL

  50. To add vectors numerically, and their components. If u = (x ,y ) and v = (x ,y ), then u + v =(x +x , y +y ). 1 1 2 2 1 2 1 2 CONFIDENTIAL

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