1. Total degrees in a triangle:2. Three angles of the triangle below:3. Three sides of the triangle below:4. Pythagorean Theorem:

x2 + y2 = r2

180

A

B

C

x, y, and r

y

x

r

HYPOTENUSE

A, B, and C

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

Q

y

x

r

sin Q = =opp. y hyp. r

cos Q = =adj. x hyp. r

tan Q = =opp. y adj. x

sin A = opposite

hypotenuse

cos A = adjacent

hypotenuse

tan A = opposite adjacentsin A = 1

2

cos A =

tan A =

√3 2

12

3A

B

C

1 √3

For <A below, calculate Sine, Cosine, and Tangent:

ac

A

B

Cb

Law of Cosines:c2 = a2 + b2 – 2ab cos C

Law of Sines:sin A sin B sin C a b c

= =

1. Scalar – a variable whose value is expressed only as a magnitude or quantityHeight, pressure, speed, density, etc.

2. Vector – a variable whose value is expressed both as a magnitude and directionDisplacement, force, velocity, momentum, etc.

3. Tensor – a variable whose values are collections of vectors, such as stress on a material, the curvature of space-time (General Theory of Relativity), gyroscopic motion, etc.

Properties of Vectors

1. MagnitudeLength implies magnitude of vector

2. DirectionArrow implies direction of vector

3. Act along the line of their direction4. No fixed origin

Can be located anywhere in space

Magnitude, Direction

Vectors - Description

45o40 lb

s

F = 40 lbs 45o

F = 40 lbs @ 45o

magnitude direction

Hat signifies vector quantity

Bold type and an underline F also identify vectors

1. We can multiply any vector by a whole number.2. Original direction is maintained, new magnitude.

Vectors – Scalar Multiplication

2

½

1. We can add two or more vectors together. 2. 2 methods:

1. Graphical Addition/subtraction – redraw vectors head-to-tail, then draw the resultant vector. (head-to-tail order does not matter)

Vectors – Addition

Vectors – Rectangular Components

y

x

F

Fx

Fy

1. It is often useful to break a vector into horizontal and vertical components (rectangular components).

2. Consider the Force vector below. 3. Plot this vector on x-y axis.4. Project the vector onto x and y axes.

Vectors – Rectangular Components

y

x

F

Fx

Fy

This means:

vector F = vector Fx + vector Fy

Remember the addition of vectors:

Vectors – Rectangular Components

y

x

F

Fx

Fy

Fx = Fx i

Vector Fx = Magnitude Fx times vector i

Vector Fy = Magnitude Fy times vector j

Fy = Fy j

F = Fx i + Fy j

i denotes vector in x direction

j denotes vector in y direction

Unit vector

Vectors – Rectangular Components

y

x

F

Fx

Fy

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

Vectors – Rectangular Components

If vector

V = a i + b j + c k

then the magnitude of vector V

|V| =

Vectors – Rectangular Components

F

Fx

Fy

cos Q = Fx / F

Fx = F cos Qi

sin Q = Fy / F

Fy = F sin Qj

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

Q

Vectors – Rectangular Components

y

x

F Fx +

Fy +

When are Fx and Fy Positive/Negative?

FFx -

Fy +

FFFx -Fy -

Fx +Fy -

Vectors – Rectangular Components

Complete the following chart in your notebook:

III

III IV

1. Vectors can be completely represented in two ways:1. Graphically2. Sum of vectors in any three independent directions

2. Vectors can also be added/subtracted in either of those ways:1.

2. F1 = ai + bj + ck; F2 = si + tj + uk

F1 + F2 = (a + s)i + (b + t)j + (c + u)k

Vectors

A third way to add, subtract, and otherwise decompose vectors:

Use the law of sines or the law of cosines to find R.

Vectors

F1 F2

R45o

105o

30o

Brief note about subtraction1. If F = ai + bj + ck, then – F = – ai – bj – ck

2. Also, if

F =

Then,

– F =

Vectors

Resultant Forces

Resultant forces are the overall combination of all forces acting on a body.

1) find sum of forces in x-direction

2) find sum of forces in y-direction

3) find sum of forces in z-direction

3) Write as single vector in rectangular components

R = SFxi + SFyj + SFzk

Resultant Forces – Example 1A satellite flies without friction in space. Earth’s gravity pulls downward on the satellite with a force of 200 N. Stray space junk hits the satellite with a force of 1000 N at 60o to the horizontal. What is the resultant force acting on the satellite?

1. Sketch and label free-body diagram (all external and reactive forces acting on the body)

2. Decompose all vectors into rectangular components (x, y, z)

3. Add vectors

Example 2

A stop light is held by two cables as shown. If the stop light weighs 120 N, what are the tensions in the two cables?