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LESSON 8 –7

LESSON 8 –7. Vectors. Five-Minute Check (over Lesson 8–6) TEKS Then/Now New Vocabulary Example 1: Represent Vectors Geometrically Key Concept: Vector Addition Example 2: Find the Resultant of Two Vectors Example 3: Write a Vector in Component Form

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LESSON 8 –7

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  1. LESSON 8–7 Vectors

  2. Five-Minute Check (over Lesson 8–6) TEKS Then/Now New Vocabulary Example 1: Represent Vectors Geometrically Key Concept: Vector Addition Example 2: Find the Resultant of Two Vectors Example 3: Write a Vector in Component Form Example 4: Find the Magnitude and Direction of a Vector Key Concept: Vector Operations Example 5: Operations with Vectors Example 6: Real-World Example: Vector Applications Lesson Menu

  3. Find s if the measures of ΔRST are mR = 63, mS = 38, and r = 52. A. 50.1 B. 44.6 C. 39.3 D. 35.9 5-Minute Check 1

  4. Find mRif the measures of ΔRST are mS = 122, s = 10.8, and r = 5.2. A. 21.3 B. 24.1 C. 29 D. 58 5-Minute Check 2

  5. Use the measures of ΔABC to find c to the nearest tenth. A. 12.7 B. 10.8 C. 9.62 D. 8.77 5-Minute Check 3

  6. Use the measures of ΔABC to find mB to the nearest degree. A. 21° B. 19° C. 18° D. 16° 5-Minute Check 4

  7. On her delivery route, Gina drives 15 miles west, then makes a 68° turn and drives southeast 14 miles. When she stops, approximately how far from her starting point is she? A. 21 mi B. 18 mi C. 16 mi D. 15.5 mi 5-Minute Check 5

  8. Targeted TEKS G.9(A) Determine the lengths of sides and measures of angles in a right triangle by applying the trigonometric ratios sine, cosine, and tangent to solve problems. Mathematical Processes G.1(B), G.1(E) TEKS

  9. You used trigonometry to find side lengths and angle measures of right triangles. • Perform vector operations geometrically. • Perform vector operations on the coordinate plane. Then/Now

  10. vector • magnitude • direction • resultant • parallelogram method • triangle method • standard position • component form Vocabulary

  11. A. Use a ruler and a protractor to draw each vector. Include a scale on each diagram. = 80 meters at 24° west of north Represent Vectors Geometrically Using a scale of 1 cm : 50 m, draw and label an 80 ÷ 50 or 1.6-centimeter arrow 24º west of the north-south line on the north side. Answer: Example 1

  12. B. Use a ruler and a protractor to draw each vector. Include a scale on each diagram. = 16 yards per second at 165° to the horizontal Represent Vectors Geometrically Using a scale of 1 cm : 8 yd/s, draw and label a 16 ÷ 8 or 2-centimeter arrow at a 165º angle to the horizontal. Answer: Example 1

  13. Using a ruler and a protractor, draw a vector to represent feet per second 25 east of north. Include a scale on your diagram. A. B. C. D. Example 1

  14. Concept

  15. Copy the vectors. Then find b a Find the Resultant of Two Vectors Subtracting a vector is equivalent to adding its opposite. Example 2

  16. Step 1 , and translate it so that its tail touches the tail of . a a –b –b Find the Resultant of Two Vectors Method 1 Use the parallelogram method. Example 2

  17. a – b a –b Find the Resultant of Two Vectors Step 2 Complete the parallelogram. Then draw the diagonal. Example 2

  18. Step 1 , and translate it so that its tail touches the tail of . –b a Find the Resultant of Two Vectors Method 2 Use the triangle method. Example 2

  19. Step 2 Draw the resultant vector from the tail of to the tip of – . –b a a – b a – b Find the Resultant of Two Vectors Answer: Example 2

  20. Copy the vectors. Then find a b A. B. C.D. a – b a – b a – b a – b Example 2

  21. Write the component form of . Write a Vector in Component Form Example 3

  22. Write a Vector in Component Form Find the change of x-values and the corresponding change in y-values. Component form of vector Simplify. Example 3

  23. Write the component form of . A. B. C. D. Example 3

  24. Find the magnitude and direction of Distance Formula (x1, y1) = (0, 0) and (x2, y2) = (7, –5) Find the Magnitude and Direction of a Vector Step 1 Use the Distance Formula to find the vector’s magnitude. Simplify. Use a calculator. Example 4

  25. Graph , its horizontal component, and its vertical component. Then use the inverse tangent function to find θ. Find the Magnitude and Direction of a Vector Step 2 Use trigonometry to find the vector’s direction. Example 4

  26. Definition of inverse tangent Use a calculator. The direction of is the measure of the angle that it makes with the positive x-axis, which is about 360 – 35.5 or 324.5. So, the magnitude of is about 8.6 units and the direction is at an angle of about 324.5º to the horizontal. Answer: Find the Magnitude and Direction of a Vector Example 4

  27. Find the magnitude and direction of A. 4; 45° B. 5.7; 45° C. 5.7; 225° D. 8; 135° Example 4

  28. Concept

  29. Find each of the following for and . Check your answers graphically.A. Operations with Vectors Check Graphically Solve Algebraically Example 5

  30. Find each of the following for and . Check your answers graphically.B. Operations with Vectors Check Graphically Solve Algebraically Example 5

  31. Find each of the following for and . Check your answers graphically.C. Operations with Vectors Check Graphically Solve Algebraically Example 5

  32. A. B. C. D. Example 5

  33. Draw a diagram. Let represent the resultant vector. Vector Applications CANOEING Suppose a person is canoeing due east across a river at 4 miles per hour. If the river is flowing south at 3 miles per hour, what is the resultant speed and direction of the canoe? Example 6

  34. Vector Applications The component form of the vector representing the velocity of the canoe is 4, 0, and the component form of the vector representing the velocity of the river is 0, –3. The resultant vector is 4, 0 + 0, –3 or 4, –3, which represents the resultant velocity of the canoe. Its magnitude represents the resultant speed. Example 6

  35. Vector Applications Use the Distance Formula to find the resultant speed. Distance Formula (x1, y1) = (0, 0) and (x2, y2) = (4, –3) The resultant speed of the canoe is 5 miles per hour. Example 6

  36. Vector Applications Use trigonometry to find the resultant direction. Definition of inverse tangent Use a calculator. The resultant direction of the canoe is about 36.9° south of due east. Answer: Therefore, the resultant speed of the canoe is 5 mile per hour at an angle of about 90° – 36.9° or 53.1° east of south. Example 6

  37. KAYAKING Suppose a person is kayaking due east across a lake at 7 miles per hour. If the lake is flowing south at 4 miles an hour, what is the resultant direction and speed of the canoe? A. Direction is about 60.3° south of due east with a velocity of about 8.1 miles per hour. B. Direction is about 60.3° south of due east with a velocity of about 11 miles per hour. C. Direction is about 29.7° south of due east with a velocity of about 8.1 miles per hour. D. Direction is about 29.7° south of due east with a velocity of about 11 miles per hour. Example 6

  38. LESSON 8–7 Vectors

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