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

6.1 Law of Sines

Deriving the Law of SinesSolving ASA and AAS casesSolving the ambiguous SSA case

The Law of Sines

Using the Law of Sines (ASA case)

Example: Solve this triangle:Solution:

º

= 180º - (45.1º + 75.8º) = 59.1º

.42.81.59sin

1.45sin2.10

1.45sin2.101.59sin2.10

1.59sin1.45sin

ina

aa

min. 18 hr. 1about or hr, 3.1mi/hr 130

mi 164

mi.164

12cos)258)(5.97(2)258()5.97(

mi. 97.5 hr. 3/4mi./hr. 130222

d

d

Using the Law of Sines (AAS case)

Example: Solve this triangle:º - (63º + 38º) = 79º

.1363sin

79sin12

79sin1263sin

79sin

12

63sin

inc

cc

SSA Variations

6.2 Law of Cosines

Deriving the Law of CosinesSolving the SAS caseSolving the SSS case

Law of Cosines

Strategy for Solving the SAS Case

Using the Law of Cosines (SAS case)

'40100)45'2034(180

4598.7

'2034sin10sin

98.7

'2034sin

10

sin

mb 98.7'2034cos)10)(9.13(2)10()9.13(

. and , b,for triangle thisSolve :Example

22

Strategy for the SSS Case

Navigation

min. 18 hr. 1about or hr, 3.1mi/hr 130

mi 164

mi.16412cos)258)(5.97(2)258()5.97(

mi. 97.5 hr. 3/4mi./hr. 130222

dd

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 givenTwo sides and included angle givenThree 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 m2

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 = (12 + 8 + 6)/2 = 13 cm.

A = √(13(13-12)(13-8)(13-6) =

√(13(1)(5)(7) ≈ 21 cm2

6.4 Vectors

Velocity and standard vectorsVector addition and Scalar multiplicationAlgebraic PropertiesVelocity VectorsForce VectorsStatic Equilibrium

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

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

10

1,

10

3)1,3(

10

1

||

1

1013||)1,3(

v.ofdirection in r unit vecto Find :Example

22

vv

u

vv

Algebraic Properties of Vectors

The Dot Product

The dot product of two vectorsAngle between two vectorsScalar component of one vector onto

anotherWork

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

3.421713

11cos

1713

11cos

)1,4(),3,2(

1

vu

Scalar Component of u on v

67.217

11

)1,4(),3,2(

u

vu

vcomp

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