4-1 Properties of Vectors

The line and arrow used in Ch.3 showing
magnitude and direction is known as the
Graphical Representation
◦ Used when drawing vector diagrams
When using printed materials, it is known as
Algebraic Representation
◦ Italicized letter in boldface
◦ d = 50 km SW
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
This property of vectors makes it possible to move
vectors graphically for adding or subtracting
Vectors shown are unequal, even though they
start at the same place
◦ C
The resultant vector is the displacement of
the vector additions.
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
When manipulating graphical reps. of vectors,
need a ruler to measure correct length
Take the tail end and place at the head of the
◦ 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
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
Find the magnitude of the sum of a 15 km
displacement and a 25 km displacement
when the angle between them is 135o.
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
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?
A person walked 450.0 m North. The person
then turned left 65o and traveled 250.0
meters. Find the resultant vector.
Multiplying a Vector by a scalar number
changes its length, but not direction, unless
◦ Vector direction is then reversed
◦ To subtract two vectors, reverse direction of the 2nd
vector then add them
◦ 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
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
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
◦ vbus relative to the street
◦ vyou relative to the bus
◦ vyou relative to the street
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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