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
