### 4-1 Properties of Vectors

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The line and arrow used in Ch.3 showing
magnitude and direction is known as the
Graphical Representation
◦ Used when drawing vector diagrams
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When using printed materials, it is known as
Algebraic Representation
◦ Italicized letter in boldface
◦ d = 50 km SW
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Two displacements are equal when the two
distances and directions are equal
◦ A and B are equal, even though they don’t begin or
end at the same place
A
This property of vectors makes it possible to move
vectors graphically for adding or subtracting
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Vectors shown are unequal, even though they
start at the same place
◦ C
D
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The resultant vector is the displacement of
My route to school is
My resultant vector is R
0.50 miles East
2.0 miles North
2.5 miles East
20.0 miles North
2.5 miles East
Resultant Vector = 23 miles NE
R
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When manipulating graphical reps. of vectors,
need a ruler to measure correct length
Take the tail end and place at the head of the
arrow
◦ Enroute to a school, someone travels 1.0 km W, 2.0
km S, and then 3.0 km W
◦ Resultant vector =
 4.5 km SW
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Vectors added at right angels can use the
Pythagorean System to find magnitude
If vectors added and angle is something other
than 90o, use the Law of Cosines
◦ R2 = A2 + B2 – 2ABcosθ
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Find the magnitude of the sum of a 15 km
displacement and a 25 km displacement
when the angle between them is 135o.
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A = 15 km; B = 25 km; θ = 135o; R = unknown
R2 = A2 + B2 – 2ABcosθ
= (25 km)2 + (15 km)2 – 2(25km)(15 km)cos135o
=625 km2 + 225 km2 – 750km2(-0.707)
=1380 km2
R = √1380km2
= 37 km
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A hiker walks 4.5 km in one direction, then makes
a 45o turn to the right and walks another 6.4 km.
What is the magnitude of her displacement?
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A person walked 450.0 m North. The person
then turned left 65o and traveled 250.0
meters. Find the resultant vector.
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Multiplying a Vector by a scalar number
changes its length, but not direction, unless
negative
◦ Vector direction is then reversed
◦ To subtract two vectors, reverse direction of the 2nd
◦ Δv = v2 – v1
◦ Δv = v2 + (-v1)
◦ If v1 is multiplied by -1, the direction of v1 is
reversed and can be added to v2 to get Δv
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Graphical addition can be used when solving
problems that involve relative velocity
◦ School bus traveling at a velocity of 8 m/s. You
walk toward the front at 3 m/s. How fast are you
moving relative to the street?
◦ vbus relative to street
◦ vyou relative to bus
◦ vyou relative to the street
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When a coordinate system is moving, two velocities
add if both moving in the same direction & subtract
if the motions are in opposite directions
◦ What if you use the same velocities and walk to the rear of
the bus. What is your resultant velocity relative to the
street?
◦ vbus relative to the street
◦ vyou relative to the bus
◦ vyou relative to the street
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Suppose an airplane pilot wants to fly from
the U.S. to Europe. Does the pilot aim the
plane straight to Europe?
◦ No, must take in consideration for wind velocity
 v
 v
air relative to the ground
plane relative to air
 v
plane relative to ground
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