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Chapter 6 Additional Topics: Triangles and Vectors

Chapter 6 Additional Topics: Triangles and Vectors. 6.1 Law of Sines 6.2 Law of Cosines 6.3 Areas of Triangles 6.4 Vectors 6.5 The Dot Product. 6.1 Law of Sines. Deriving the Law of Sines Solving ASA and AAS cases Solving the ambiguous SSA case. The Law of Sines.

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Chapter 6 Additional Topics: Triangles and Vectors

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  1. Chapter 6Additional Topics: Triangles and Vectors 6.1 Law of Sines 6.2 Law of Cosines 6.3 Areas of Triangles 6.4 Vectors 6.5 The Dot Product

  2. 6.1 Law of Sines • Deriving the Law of Sines • Solving ASA and AAS cases • Solving the ambiguous SSA case

  3. The Law of Sines

  4. Using the Law of Sines (ASA case) • Example: Solve this triangle: • Solution: a + b + g = 180º g = 180º - (45.1º + 75.8º) = 59.1º

  5. Using the Law of Sines (AAS case) • Example: Solve this triangle: g = 180º - (63º + 38º) = 79º

  6. SSA Variations

  7. 6.2 Law of Cosines • Deriving the Law of Cosines • Solving the SAS case • Solving the SSS case

  8. Law of Cosines

  9. Strategy for Solving the SAS Case

  10. Using the Law of Cosines (SAS case)

  11. Strategy for the SSS Case

  12. Navigation • Example: Find how far a plane has flown off course at 12º after flying for ¾ of an hour. • Also, find how much longer the flight will take.

  13. 6.3 Area of Triangles • Base and height given • Two sides and included angle given • Three sides given (Heron’s Formula) • Arbitrary triangles

  14. Base and Height Given • Example: Find the area of this triangle. • Solution: • A = (ab/2) sin q = ½ (8m)(5m) sin 35º ≈ 11.5 m2

  15. Three Sides Given

  16. Using Heron’s Formula • Example: Find the area of the triangle with sides a = 12 cm, b = 8 cm, and c = 6 cm. • Solution: s = (12 + 8 + 6)/2 = 13 cm. A = √(13(13-12)(13-8)(13-6) = √(13(1)(5)(7) ≈ 21 cm2

  17. 6.4 Vectors • Velocity and standard vectors • Vector addition and Scalar multiplication • Algebraic Properties • Velocity Vectors • Force Vectors • Static Equilibrium

  18. Finding a Standard Vector for a Given Geometric Vector • The coordinates (x, y) of P are given by x = xb – xa = 4 – 8 = -4 y = yb – ya = 5 – (-3) = 8

  19. Vector Addition

  20. Scalar Multiplication • Let u = (-5, 3) and v = (4, -6) • u + v = (-5 + 4, 3 + (-6)) = (-1, -3) • -3 u = -3(-5, 3) = (-3(-5), -3(3)) = (15, -9)

  21. Unit Vectors

  22. Algebraic Properties of Vectors

  23. The Dot Product • The dot product of two vectors • Angle between two vectors • Scalar component of one vector onto another • Work

  24. The Dot Product

  25. Computing Dot Products • Example: Find the dot product of (4,2) and (1,-3) • Solution: (4,2)·(1,-3)=4·1 + 2·(-3) = -2

  26. Angle Between Two Vectors

  27. Scalar Component of u on v

  28. Work • Example: How much work is done by a force F = (6,4) that moves an object from the origin to the point p = (8, 2)? • Solution: w = (6,4)·(8,2) = 56 ft-lb

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