Vectors
Vectors and Scalars
• Vector: Quantity which requires both
magnitude (size) and direction to be
completely specified
– 2 m, west; 50 mi/h, 220o
– Displacement; Velocity
• Scalar: Quantity which is specified
completely by magnitude (size)
– 2 m; 50 mi/h
– Distance; Speed
Vector Representation
• Print notation: A
– Sometimes a vector is
indicated by printing
the letter representing
the vector in bold face
Mathematical Reference System
90o
y
180o
Angle is measured
counterclockwise
wrt positive x-axis
x
270o
0o
Equal and Negative Vectors
Vector Addition
A + B = C (head to tail method)
B + A = C (head to tail method)
A + B = C (parallelogram method)
Addition of Collinear Vectors
Adding Three Vectors
Vector Addition Applets
• Visual Head to Tail Addition
• Vector Addition Calculator
Subtracting Vectors
Vector Components
Vertical Component
Ay= A sin 
Horizontal Component
Ax= A cos 
Signs of Components
Components ACT
• For the following, make a sketch and then
resolve the vector into x and y components.
A   60 m,120
o

Ay
Ax
B   40 m, 225
o

Bx
By
Ax = (60 m) cos(120) = -30 m
Bx = (40 m) cos(225) = -28.3 m
Ay = (60 m) sin(120) = 52 m
By = (40 m) sin(225) = -28.3 m
(x,y) to (R,)

D  Dx  Dy
2
 Dy 

  tan 1 
 Dx 


2
• Sketch the x and y components
in the proper direction
emanating from the origin of the
coordinate system.
• Use the Pythagorean theorem to
compute the magnitude.
• Use the absolute values of the
components to compute angle  - the acute angle the resultant
makes with the x-axis
• Calculate  based on the
quadrant*
  360  
o
*Calculating θ
• When calculating the angle,
• 1) Use the absolute values of the
components to calculate 
• 2) Compute C using inverse tangent
• 3) Compute  from  based on the quadrant.
• Quadrant I:  = 
• Quadrant II:  = 180o - ;
• Quadrant III:  = 180o + 
• Quadrant IV:  = 360o - 
(x,y) to (R,) ACT
• Express the vector in (R,) notation
(magnitude and direction)
A = (12 cm, -16 cm)
A = (20 cm, 307o)
Vector Addition by
Components
• Resolve the vectors
into x and y
components.
• Add the x-components
together.
• Add the y-components
together.
• Use the method shown
previously to convert
the resultant from
(x,y) notation to (R,)
notation
Practice Problem
Given A = (20 m, 40o) and B = (30 m, 100o), find the
vector sum A + B.
A = (15.32 m, 12.86 m)
B = (-5.21 m, 29.54 m)
A + B = (10.11 m, 42.40 m)
A + B = (43.6 m, 76.6o)
Unit Vectors: x̂ ŷ Notation
• Vector A can be
expressed in several ways
• Magnitude & Direction
(A,)
• Rectangular Components
ŷ
ŷ
x̂
x̂
(Ax , AY)
Ax xˆ  Ay yˆ