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

Using the Law of Sines (ASA case) Example: Solve this triangle: Solution:  º  = 180º - (45.1º º) = 59.1º

Using the Law of Sines (AAS case) Example: Solve this triangle:  º - (63º + 38º) = 79º

SSA Variations

6.2 Law of Cosines Deriving the Law of Cosines Solving the SAS case Solving the SSS case

Law of Cosines

Strategy for Solving the SAS Case

Using the Law of Cosines (SAS case)

Strategy for the SSS Case

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.

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

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

Three Sides Given

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 = ( )/2 = 13 cm. A = √(13(13-12)(13-8)(13-6) = √(13(1)(5)(7) ≈ 21 cm 2

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

Finding a Standard Vector for a Given Geometric Vector The coordinates (x, y) of P are given by x = x b – x a = 4 – 8 = -4 y = y b – y a = 5 – (-3) = 8

Vector Addition

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)

Unit Vectors

Algebraic Properties of Vectors

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

The Dot Product

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

Angle Between Two Vectors

Scalar Component of u on v

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