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Vectors

1. 2. 3. Vectors. Lesson 8-6. Check Skills You’ll Need. (For help, go to Lesson 8-1.). Find the value of x. Check Skills You’ll Need. 8-6. 10 65. Vectors. Lesson 8-6. Check Skills You’ll Need. Solutions.

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Vectors

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  1. 1. 2. 3. Vectors Lesson 8-6 Check Skills You’ll Need (For help, go to Lesson 8-1.) Find the value of x. Check Skills You’ll Need 8-6

  2. 10 65 Vectors Lesson 8-6 Check Skills You’ll Need Solutions 1. Use the Pythagorean Theorem: a2 + b2 = c2 (16)2 + (20)2 = x2 256 + 400 = x2x2 = 656 x = 656 = 4 41 2. Use the Pythagorean Theorem: a2 + b2 = c2x2 + (6)2 = (7)2 x2 + 36 = 49 x2 = 49 – 36 = 13 x = 13 3. Use the Pythagorean Theorem: a2 + b2 = c2 402 + 702 = x2 1600 + 4900 = x2 6500 = x2x = 6500 = 8-6

  3. Angles of Elevation and Depression Lesson 8-5 Lesson Quiz Use the diagram for Exercises 1 and 2. 1. Describe how 1 relates to the situation. 2. Describe how 2 relates to the situation. angle of elevation from man’s eyes to treetop angle of depression from treetop to man’s eyes A 6-ft man stands 12 ft from the base of a tree. The angle of elevation from his eyes to the top of the tree is 76°. 3. About how tall is the tree? 4. If the man releases a pigeon that flies directly to the top of the tree, about how far will it fly? 5. What is the angle of depression from the treetop to the man’s eyes? about 54 ft about 50 ft 76° 8-6

  4. Textbook

  5. You can think of a vector as a directed line segment. The vector below may be named Vectors Lesson 8-6 Notes The speed and direction an object moves can be represented by a vector. A vectoris a quantity that has both length and direction. 8-6

  6. A vector can also be named using component form. The component form<x, y> of a vector lists the horizontaland verticalchange from the initial point to the terminal point. The component form of is <2, 3>. Vectors Lesson 8-6 Notes 8-6

  7. The magnitudeof a vector is its length. The magnitude of a vector is written Vectors Lesson 8-6 Notes 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. 8-6

  8. The directionof 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 is 60°. The direction of a vector can also be given as a bearing relative to the compass directions north, south, east, and west. has a bearing of 30° East of North. Vectors Lesson 8-6 Notes 8-6

  9. Describe OM as an ordered pair. Give coordinates to the nearest tenth. x 80 y 80 cos 40° = Use sine and cosine. sin 40° = x = 80(cos 40°) Solve for the variable.y = 80(sin 40°) x 61.28355545 Use a calculator.y 51.42300877 Because point M is in the third quadrant, both coordinates are negative. To the nearest tenth, OM –61.3,–51.4 . Vectors Lesson 8-6 Additional Examples Describing a Vector Use the sine and cosine ratios to find the values of x and y. Quick Check 8-6

  10. Vectors Lesson 8-6 Additional Examples Describing a Vector Direction Use compass directions to describe the direction of the vector. The angle is measured from due south toward east. Because the vector forms a 22° angle with the south segment, it is 22° east of south. Quick Check 8-6

  11. Draw a diagram to represent the situation. d = (12 – 0)2 + (–9 – 0)2 Distance Formula Simplify. d = 144 + 81 Simplify. d = 225 Take the square root. d = 15 Vectors Lesson 8-6 Additional Examples Real-World Connection A boat sailed 12 mi east and 9 mi south. The trip can be described by the vector  12, –9 . Use distance and direction to describe this vector a second way. To find the distance sailed, use the Distance Formula. 8-6

  12. (continued) Use the tangent ratio. 9 12 tan x° = = 0.75 x = tan–1 (0.75) Find the angle whose tangent is 0.75. 0.75 36.869898 Use a calculator. Vectors Lesson 8-6 Additional Examples To find the direction the boat sails, find the angle that the vector forms with the x-axis. The boat sailed 15 mi at about 37° south of east. Quick Check 8-6

  13. Two vectors are equal vectorsif they have the same magnitude and the same direction. For example, . Equal vectors do not have to have the same initial point and terminal point. Vectors Lesson 8-6 Notes 8-6

  14. Two vectors are parallel vectorsif they have the same direction or if they have opposite directions. They may have different magnitudes. For example, Equal vectors are always parallel vectors. Vectors Lesson 8-6 Notes 8-6

  15. Vectors Lesson 8-6 Notes The resultant vectoris 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. 8-6

  16. Vectors Lesson 8-6 Notes 8-6

  17. To add vectors numerically, add their components. If = <x1, y1> and = <x2, y2>, then = <x1 + x2, y1 + y2>. Vectors Lesson 8-6 Notes 8-6

  18. To find the first coordinate of s, add the first coordinates of v and w. To find the second coordinate of s, add the second coordinates of v and w. s =  4, 3  =  4, –3  Add the coordinates. =  4 + 4, 3 + (–3)  Simplify. =  8, 0  Vectors Lesson 8-6 Additional Examples Adding Vectors Vectors v 4, 3  and w 4, –3  are shown below. Write s, their sum, as an ordered pair. Quick Check 8-6

  19. Draw a diagram to represent the situation. c2 = 2502 + 202 The lengths of the legs are 250 and 20. c2 = 62,900 Simplify. c250.798724 Use a calculator. Vectors Lesson 8-6 Additional Examples Real-World Connection An airplane’s speed is 250 mi/h in still air. The wind is blowing due east at 20 mi/h. If the airplane heads directly north, what is its resultant speed and direction? The diagram shows the sum of the two vectors. To find the resultant speed, use the Pythagorean Theorem. 8-6

  20. 20 250 Use the tangent ratio. tan x° = = 0.08 x = tan–1 (0.08) Use the inverse of the tangent. Use a calculator. x4.573921 Vectors Lesson 8-6 Additional Examples Quick Check (continued) To find the direction of the airplane’s flight, use trigonometry to find x. The airplane’s speed is about 251 mi/h. Its direction is about 5° east of north. 8-6

  21. 1. Describe the vector as an ordered pair. Round coordinates to the nearest tenth. 2. Use compass directions to describe the direction of ON. 4. Write the vector v = a + b as an ordered pair. Vectors Lesson 8-6 Lesson Quiz Use the diagram for Exercises 1 and 2.  46.4, 18.7  22° north of east 3. Iris rode her bike 30 mi south and 16 mi west of her home. Her trip can be described by the vector  –16, –30 . Use distance and direction to describe the vector a second way. Iris rode 34 mi at about 28° west of south. 5, 2 5. An airplane has a speed of 240 mi/h in still air. The plane heads due north and encounters a 30-mi/h wind blowing due east. Find the resultant speed and direction. Round to the nearest unit. 242 mi/h at 7° east of north 8-6

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