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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
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Mathematical Reference System
90o
y
180o
Angle is measured
counterclockwise
wrt positive x-axis
x
0o
270o
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)
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Addition of Collinear Vectors
Adding Three Vectors
Vector Addition Applets
• Visual Head to Tail Addition
• Vector Addition Calculator
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Subtracting Vectors
Vector Components
Vertical Component
Ay= A sin θ
Horizontal Component
Ax= A cos θ
Signs of Components
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Components ACT
• For the following, make a sketch and then
resolve the vector into x and y components.
A = ( 60 m,120o )
B = ( 40 m, 225o )
(x,y) to (R,θ)
θθ
D = Dx 2 + Dy 2
 Dy
 Dx

α = tan −1 




• 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 α - the acute angle the resultant
makes with the x-axis
• Calculate θ based on the
quadrant*
θ = 360o − α
*Calculating θ
• When calculating the angle,
• 1) Use the absolute values of the
components to calculate α
• 2) Compute C using inverse tangent
• 3) Compute θ from α based on the quadrant.
• Quadrant I: θ = α
• Quadrant II: θ = 180o - α
• Quadrant III: θ = 180o + α
• Quadrant IV: θ = 360o - α
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(x,y) to (R,θ) ACT
• Express the vector in (R,θ) notation
(magnitude and direction)
A = (12 cm, -16 cm)
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,θ)
notation
Practice Problem
Given A = (20 m, 40o) and B = (30 m, 100o), find the
vector sum A + B.
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Unit Vectors: x̂ ŷ Notation
• Vector A can be
expressed in several ways
• Magnitude & Direction
(A,θ)
• Rectangular Components
ŷ
ŷ
x̂
x̂
(Ax , AY)
Ax xˆ + Ay yˆ
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