Principles Of Engineering

© 2012 Project Lead The Way, Inc.

Vectors

Vector Quantities

Have both a magnitude and direction

Examples: Position, force, moment

Vector Notation

Vectors are given a variable, such as A or B

Handwritten notation usually includes an r ur arrow, such as u

A or B

Illustrating Vectors

Vectors are represented by arrows

Include magnitude, direction, and sense

Magnitude: The length of the line segment

Magnitude = 3

30°

+X

Illustrating Vectors

Vectors are represented by arrows

Include magnitude, direction, and sense

Direction: The angle between a reference axis and the arrow’s line of action

Direction = 30° counterclockwise from the positive x-axis

30°

+x

Illustrating Vectors

Vectors are represented by arrows

Include magnitude, direction, and sense

Sense: Indicated by the direction of the tip of the arrow

Sense = Upward and to the right

30°

+x

Sense

+y (up)

+y (up)

-x (left)

+x (right)

(0,0)

-y (down)

-x (left)

-y (down)

+x (right)

Trigonometry Review

Right Triangle

A triangle with a 90° angle

Sum of all interior angles = 180°

Pythagorean Theorem: c2 = a2 + b2

H

us n te o p y p) y h e(

Opposite Side

(opp)

90°

Adjacent Side (adj)

Trigonometry Review

Trigonometric Functions soh cah toa sin θ° = opp / hyp cos θ° = adj / hyp tan θ° = opp / adj

H

us n te o p y p) y h e(

Opposite Side

(opp)

90°

Adjacent Side (adj)

Trigonometry Application

The hypotenuse is the Magnitude of the

Force, F

In the figure here,

The adjacent side is the x-component, Fx

The opposite side is the y-component, Fy

H

se u ten o yp

F

Opposite Side

Fy

90°

Adjacent Side Fx

Trigonometry Application sin θ° = Fy / F

Fy= F sin θ°

cos θ° = Fx / F

Fx= F cos θ°

tan θ° = Fy / Fx

H

ten o yp

use

F

Opposite Side

Fy

90°

Adjacent Side Fx

Fx and Fy are negative if left or down, respectively.

VectorurX and Y Components

Vector A

Magnitude = 75.0 lb

Direction = 35.0°CCW from positive x-axis

+y

Sense = right, up ur A 75.0 lb opp = FAy

35.0°

-x

-y

adj = FAx

+x

Vector X and Y Components

Solve for FAx

FAx

cos35.0

75.0 lb

adj cos hyp FAx 75.0 lb cos 35.0 up

+y

ur

A 75.0 lb

FAx 61.4 lb opp = FAy

35.0°

-x

adj = FAx

-y

+x

Vector X and Y Components

Solve for FAy

FAy

opp sin hyp F sin ur

A

+y

FAY 75.0 lb sin 35.0 up

Ay

sin 35.0

FAy 43.0 lb

ur

A 75.0 lb opp = FAy

35.0°

-x

adj = FAx

-y

75.0 lb

+x

Vector X and Y Components – Your Turn ur Vector B

Magnitude =

Direction =

+y

Sense = adj = FBx

-x

75.0 lb

35.0°CW from positive x-axis

right, down

+x

35.0° opp = FBy

-y

ur

B 75.0 lb

Vector X and Y Components – Your Turn

Solve for FBx adj cos hyp FBx cos35.0

75.0 lb

+y

FBx 75.0 lb cos 35.0 right adj = FBx

-x

+x

35.0° opp = FBy

-y

ur

B 75.0 lb

FBx 61.4 lb

Vector X and Y Components – Your Turn

Solve for FBY opp sin hyp sin 35.0

adj = FBx

+x

35.0° opp = FBy

-y

75.0 lb

FBy 75.0 lb sin 35.0 down

+y

-x

FBy

ur

B 75.0 lb

FBy 43.0 lb

Resultant Force

Two people are pulling a boat to shore.

They are pulling with the same magnitude. ur A 75.0 lb

35.0

35.0

ur

B 75.0 lb

Resultant Force ur A 75 lb

FAy = 43.0 lb

35

FAx = 61.4 lb

35

FBx = 61.4 lb

List the forces according to sense. Label right and up forces as positive, and label left and down forces as negative. Fx

FAx = +61.4 lb

FBx = +61.4 lb

Fy

FBy= -43.0 lb

ur

B 75 lb

FAy = +43.0 lb

FBy = -43.0 lb

Resultant Force

Sum () the forces

Fx

FAx = +61.4 lb

Fx = FAx + FBx

FBx = +61.4 lb

Fx = 61.436 lb + 61.436 lb

Fx = 122.9 lb (right)

Fy

FAy = +43.0 lb

FBy = -43.0 lb

Fy = FAy + FBy

Fy = 43.018 lb + (-43.018 lb) = 0

Magnitude is 122.9 lb

Direction is…