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Trigonometry and Vectors

REVIEW

Common triangles in Geometry and Trigonometry

You must memorize these triangles

45o

60o

2

1

1

30o

45o

1

2

3

Trigonometry and Vectors

REVIEW

Common triangles in Geometry and Trigonometry

5

3

4

1

Trigonometry and Vectors

opposite hypotenuse

sin A =

opposite adjacent

adjacent hypotenuse

tan A =

cos A =

IOTPOLYENGINEERING

3-8

REVIEW

Trigonometric Functions

Trigonometric functions are ratios of the lengths of the segments that make up angles.

Trigonometry and Vectors

IOTPOLYENGINEERING

3-9

Vectors

- Scalar Quantities – a quantity that involves magnitude only; direction is not important
- Tiger Woods – 6’1”
- Shaquille O’Neill – 7’0”

- Vector Quantities – a quantity that involves both magnitudeand direction

How hard to impact the cue ball is only part of the game – you need to know direction too

Weight is a vector quantity

Trigonometry and Vectors

IOTPOLYENGINEERING

3-9

Scalar or Vector?

- 400 mph due north
- $100
- 10 lbs weight

- 5 miles northeast
- 6 yards
- 1000 lbs force

Magnitude and Direction

Magnitude and Direction

Vector

Vector

Magnitude only

Magnitude only

Scalar

Scalar

Magnitude and Direction

Magnitude only

Vector

Scalar

Trigonometry and Vectors

IOTPOLYENGINEERING

3-9

Vectors

- Free-body Diagram
- A diagram that shows all external forces acting on an object.

applied force

normal force

N

F

Ff

friction force

force of gravity (weight)

Wt

Trigonometry and Vectors

IOTPOLYENGINEERING

3-9

Vectors

- Describing vectors –
- We MUST represent both magnitudeand direction.
- Describe the force applied to the wagon by the skeleton:

Hat signifies vector quantity

40 lbs

45o

F = 40 lbs 45o

magnitude

direction

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Vectors – Scalar Multiplication

- We can multiply any vector by a real number.
- Original direction is maintained, new magnitude.

2

½

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Vectors – Addition

- We can add two or more vectors together.
- Redraw vectors head-to-tail, then draw the resultant vector.
- (head-to-tail order does not matter)

March 14, 2011

IOTPOLYENGINEERING

3-10

Drill

Find a + b

Find 2 a

y

y

a

a

b

x

x

Drill

IOTPOLYENGINEERING

3-10

Find 2 a

y

2a

a

x

Drill

IOTPOLYENGINEERING

3-10

Find a + b

y

a+b

a

a

b

x

Drill

IOTPOLYENGINEERING

3-10

Find a + b

b

y

a

a+b

b

x

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Vectors – Rectangular Components

- It is often useful to break a vector into horizontal and vertical components (rectangular components).
- Consider the Force vector below.
- Plot this vector on x-y axis.
- Project the vector onto x and y axis.

y

F

Fy

x

Fx

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Vectors – Rectangular Components

This means:

vector F = vector Fx + vector Fy

Remember the addition of vectors:

y

F

Fy

x

Fx

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Unit vector

Vectors – Rectangular Components

Vector Fx= Magnitude Fxtimes vector i

F = Fx i + Fyj

Fx= Fx i

i denotes vector in x direction

y

Vector Fy= Magnitude Fytimes vector j

F

Fy= Fy j

Fy

j denotes vector in y direction

x

Fx

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Vectors – Rectangular Components

From now on, vectors on this screen will appear as bold type without hats.

For example,

Fx = (4 lbs)i

Fy = (3 lbs)j

F = (4 lbs)i + (3 lbs)j

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Vectors – Rectangular Components

Each grid space represents 1 lb force.

What is Fx?

Fx = (4 lbs)i

What is Fy?

Fy = (3 lbs)j

What is F?

F = (4 lbs)i + (3 lbs)j

y

F

Fy

x

Fx

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Vectors – Rectangular Components

What is the relationship between q, sin q, and cos q?

cos q = Fx / F

Fx = F cos qi

sin q = Fy / F

Fy = F sin qj

F

Fy

q

Fx

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Vectors – Rectangular Components

When are Fx and Fy Positive/Negative?

Fy +

Fy +

y

F

Fx +

Fx -

F

x

F

F

Fx -

Fx +

Fy -

Fy -

IOTPOLYENGINEERING

3-10

Vectors – Rectangular Components

Complete the following chart in your notebook:

I

II

III IV

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Vectors – Rectangular Components

Each grid space represents 1 lb force.

What is Fx?

Fx = (-1 lbs)i

What is Fy?

Fy = (3 lbs)j

What is F?

F = (-1 lbs)i + (3 lbs)j

y

F

x

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Vectors – Resultant Forces

Resultant forces are the overall combination of all forces acting on a body. 1) Break up all forces into x and y component forces

2) add up all of the component forces in x-direction

3) add up all of the component forces in y-direction

4) Write resultant as single vector in rectangular components

150 lb

60o

100 lb

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

1) Break up all forces into x and y component forces

Space Junk:

150 lb

60o

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Break up all forces into x and y component forces

Gravity

100 lb

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

2) Add up all forces in x direction

150 lb

60o

100 lb

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

3) Add up all forces in y direction

150 lb

60o

100 lb

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

4) Write resultant as single vector in rectangular components

150 lb

60o

100 lb

Classwork

IOTPOLYENGINEERING

3-10

Complete Worksheet

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Vectors – Resultant Forces

Resultant forces are the overall combination of all forces acting on a body. 1) sum of forces in x-direction

2) sum of forces in y-direction

3) Write as single vector in rectangular components

Fx = F cos Qi

= (150 lbs) (cos 60) i

= (75 lbs)i

SFx= (75 lbs)i

No x-component

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Vectors – Resultant Forces

Resultant forces are the overall combination of all forces acting on a body. 1) sum of forces in x-direction

2) sum of forces in y-direction

3) Write as single vector in rectangular components

Fy = F sin Qj

= (150 lbs) (sin 60) j

= (75 lbs)j

Wy = -(100 lbs)j

SFy= (75 lbs)j - (100 lbs)j

SFy = (75 - 100 lbs)j

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Vectors – Resultant Forces

Resultant forces are the overall combination of all forces acting on a body. 1) sum of forces in x-direction

2) sum of forces in y-direction

3) Write as single vector in rectangular components

R = SFx +SFy

R = (75 lbs)i + (75 - 100 lbs)j

R = (75 lbs)i + (29.9 lbs)j

IOTPOLYENGINEERING

3-10

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Trigonometry and Vectors

CLASSWORK/ HOMEWORK

Complete problem #4 on the Vector Worksheet