Vectors
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Scalars and Vectors
• Scalar – has only magnitude (or size)
• Vector – has both magnitude and direction
Examples of Scalars:
• Mass
• Distance
• Time
• Density
• Work
• Energy
Examples of Vectors:
• Displacement
• Velocity
• Acceleration
• Force
• Momentum
• Angular momentum
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Scalars and Vectors
• Which are scalars? Which are vectors?
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Vector Addition - One Dimension
• Always add vectors “tip-to-tail”
• Resultant is the sum of the first two vectors
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• When adding vectors together,
the answer is called “The
Resultant”.
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Vector Addition – One Dimension
• Always add vectors
“tip-to-tail”
• Place the tip of the
first vector next to
the tail of the
second
• “Resultant” is the
sum of the first two
vectors
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Vector Addition – Two Dimensions
• Still add vectors “tip-to-tail”
• Resultant has both magnitude (the numerical sum of the
first two vectors) and direction (typically an angle or
direction – north, south, etc.)
• Example: A hiker travels 4 miles east
• Which is the magnitude?
• Which is the direction?
• Is this a vector?
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Vector Addition – Hiker Example
• Example: A hiker travels 8 miles east then 2 miles north.
How far is he from where he started? At what angle?
• Draw the resultant.
• Use pythagorean theorem to calculate distance.
• Use geometry (SOH-CAH-TOA) to calculate angle.
2 mi
8 mi
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SOH-CAH-TOA
• SOH: sin   opposite  y
hypotenuse
• CAH: cos  
R
adjacent
x

hypotenuse R
R
y
θ
opposite y

• TOA: tan  
adjacent x
x
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Vector Addition – Hiker Example
• Example: A hiker travels 8 miles east then 2 miles north.
How far is he from where he started? At what angle?
• Draw the resultant.
• Use pythagorean theorem to calculate distance.
• Use geometry (SOH-CAH-TOA) to calculate angle.
2 mi
8 mi
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Vector Addition
You are in a plane flying east at 45 km/hr then north at 25
km/hr. What is your resultant velocity?
45 km/hr east
+
25 km/hr north
V=?
=?
25 km/hr north
45 km/hr east
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V=?
=?
a b  c
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Vy = 25 km/hr
Vx = 45 km/hr
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2
(45km / hr ) 2  (25km / hr ) 2  V 2
2025km2 / hr 2  625km2 / hr 2  V
opp
tan  
adj
51.48km / hr  V
25km / hr
tan  
45km / hr
tan   0.556
  tan 1 0.556
 = 29.050 north of east
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Resolving Vectors
into components
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Resolving Vectors
• Component – the projection of a vector onto a
coordinate axis
• Vectors are at angles
• Want the x-component and y-component
• i.e., the x “piece” and the y “piece” of the vector
VECTOR
“Y-component”
“x-component”
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Resolving Vectors
You travel 30 meters at an angle that is 250 north of
east. Resolve this vector into its components
30 m
250

y  ?

x  ?
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30 m
250

y  ?

x  ?

x
0
cos 25 
30m

y
0
sin 25 
30m

(30m)(0.423)  y

12.68m  y

(30m)(0.906)  x

27.19m  x
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Positive or Negative Components?
y
X = neg
Y = pos
X = pos
Y = pos
x
X = neg
Y = neg
X = pos
Y = neg
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“Breaking Vectors down into
Components” Practice…
(& Walking the Vectors Lab, if time)
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