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8-6

8-6. Vectors. Warm Up. Lesson Presentation. Lesson Quiz. Holt Geometry. Warm Up Find AB. 1. A (0, 15), B (17, 0) 2. A (–4, 2), B (4, –2) Solve each equation. Round to the nearest tenth or nearest degree. 3. 4. 5. 6. 6.3. 3.1. 38°. 50°. Objectives.

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8-6

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  1. 8-6 Vectors Warm Up Lesson Presentation Lesson Quiz Holt Geometry

  2. Warm Up Find AB. 1.A(0, 15), B(17, 0) 2.A(–4, 2), B(4, –2) Solve each equation. Round to the nearest tenth or nearest degree. 3.4. 5. 6. 6.3 3.1 38° 50°

  3. Objectives Find the magnitude and direction of a vector. Use vectors and vector addition to solve real world problems.

  4. Vocabulary vector component form magnitude direction equal vectors parallel vectors resultant vector

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

  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>.

  7. So the component form of is <–3, 5>. Example 1A: Writing Vectors in Component Form Write the vector in component form. The horizontal change from H to G is –3 units. The vertical change from H to G is 5 units.

  8. with M(–8, 1) and N(2, –7) = <x2 – x1, y2 – y1> = <2 – (–8), –7 – 1> = <10, –8> Example 1B: Writing Vectors in Component Form Write the vector in component form. Subtract the coordinates of the initial point from the coordinates of the terminal point. Substitute the coordinates of the given points. Simplify.

  9. So the component form of is <–3, –4>. Check It Out! Example 1a Write the vector in component form. The horizontal change is –3 units. The vertical change is –4 units.

  10. = <x2 – x1, y2 – y1> = <6 – (–1), 2 – 1> = <7, 1> Check It Out! Example 1b Write each vector in component form. the vector with initial point L(–1, 1) and terminal point M(6, 2) Subtract the coordinates of the initial point from the coordinates of the terminal point. Substitute the coordinates of the given points. Simplify.

  11. The magnitudeof a vector is its length. The magnitude of a vector is written 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.

  12. Example 2: Finding the Magnitude of a Vector Draw the vector <–1, 5> on a coordinate plane. Find its magnitude to the nearest tenth. Step 1 Draw the vector on a coordinate plane. Use the origin as the initial point. Then (–1, 5) is the terminal point. Step 2 Find the magnitude. Use the Distance Formula.

  13. Check It Out! Example 2 Draw the vector <–3, 1> on a coordinate plane. Find its magnitude to the nearest tenth. Step 1 Draw the vector on a coordinate plane. Use the origin as the initial point. Then (–3, 1) is the terminal point. Step 2 Find the magnitude. Use the Distance Formula.

  14. 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 N 30° E.

  15. Step 2 Find the direction. Draw right triangle ABC as shown. A is the angle formed by the vector and the x-axis, and . Example 3: Finding the Direction of a Vector The force exerted by a skier is given by the vector <1, 4>. 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. B C A So mA = tan-1(4)  76°.

  16. Step 2 Find the direction. Draw right triangle ABC as shown. A is the angle formed by the vector and the x-axis, and Check It Out! Example 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. Step 1 Draw the vector on a coordinate plane. Use the origin as the initial point. B A C

  17. 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.

  18. 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.

  19. Example 4: Identifying Equal and Parallel Vectors Identify each of the following. A. equal vectors Identify vectors with the same magnitude and direction. B. parallel vectors Identify vectors with the same or opposite directions.

  20. Check It Out! Example 4 Identify each of the following. a. equal vectors Identify vectors with the same magnitude and direction. b. parallel vectors Identify vectors with the same or opposite directions.

  21. 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.

  22. To add vectors numerically, add their components. If = <x1, y1> and = <x2, y2>, then = <x1 + x2, y1 + y2>.

  23. Airplane N N Wind 400 60 70° y 20° W E W E x S S Example 5: Aviation Application An airplane is flying at a constant speed of 400 mi/h at a bearing of N 70º E. A 60 mi/h wind is blowing due north. What are the plane’s actual speed and direction? Round the speed to the nearest tenth and the direction to the nearest degree. Step 1 Sketch vectors for the airplane and the wind.

  24. Example 5 Continued Step 2 Write the vector for the airplane in component form. The airplane’s vector has a magnitude of 400 mi/h and makes an angle of 20º with the x-axis. so x = 400 cos20°  375.9. so y = 400 sin20°  136.8. The airplane’s vector is <375.9, 136.8>.

  25. Step 4 Find and sketch the resultant vector . Add the components of the airplane’s vector and the wind vector. Example 5 Continued Step 3 Write the vector for the wind in component form. Since the wind moves 60 mi/h in the direction of the y-axis, it has a horizontal component of 0 and a vertical component of 60. So its vector is <0, 60>. <375.9, 136.8> + <0, 60> = <375.9, 196.8>. The resultant vector in component form is <375.9, 196.8>.

  26. + Example 5 Continued Step 5 Find the magnitude and direction of the resultant vector. The magnitude of the resultant vector is the airplane’s actual speed (or ground speed). The angle measure formed by the resultant vector gives the airplane’s actual direction.

  27. 20° 70° Check It Out! Example 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. Step 1 Sketch vectors for the kayaker and the current.

  28. so x = 4cos70°  1.4. so y = 4 sin70°  3.8. Check It Out! Example 5 Continued Step 2 Write the vector for the kayaker in component form. The kayaker’s vector has a magnitude of 4mi/h and makes an angle of 20° with the x-axis. The kayaker’s vector is <1.4, 3.8>.

  29. Step 4 Find and sketch the resultant vector . Add the components of the kayaker’s vector and the current’s vector. <1.4, 3.8> + <1, 0> = <2.4, 3.8> The resultant vector in component form is <2.4, 3.8>. Check It Out! Example 5 Continued Step 3 Write the vector for the current in component form. Since the current moves 1 mi/h in the direction of the x-axis, it has a horizontal component of 1 and a vertical component of 0. So its vector is <1, 0>.

  30. or N 32° E Check It Out! Example 5 Continued 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.

  31. Lesson Quiz: Part I Round angles to the nearest degree and other values to the nearest tenth. 1. Write with S(–5, 2) and T(8, –4) in component form. 2. Write with magnitude 12 and direction 36° in component form. 3. Find the magnitude and direction of the vector <4, 5>. <13, –6> <9.7, 7.1> 6.4; 51°

  32. Lesson Quiz: Part II 4. Find the sum of the vectors <2, –4> and <3, 6>. Then find the magnitude and direction of the resultant vector. 5. A boat is heading due east at a constant speed of 35 mi/h. There is an 8 mi/h current moving north. What is the boat’s actual speed and direction? <5, 2>; 5.4; 22° 35.9 mi/h; N 77° E

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