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Vectors
Reference : The Glenbrook Web Site, Lesson 1 (under the Vectors section)
The Science Joy Wagon Site (under the Motion section)
Vectors are a useful concept in visually representing physical quantities that would
be difficult to represent otherwise. For us, these quantities include displacement,
velocity, acceleration, force, and momentum. Because of their concise visual nature,
we can use vectors to quickly represent and analyze these quantities in various
situations.
Vectors are quantities consisting of magnitude and direction. Some examples
of vectors that we’ll be using in this topic are:
Quantity
Example Magnitude
Example Direction
displacement
13 miles
30° West of North
velocity
8 m/sec
45° South of East
force (weight)
155 lb
down
N
2. Representation of a Vector
We represent vectors with arrows such that:
W of N
(a) The length represents the magnitude
W
(b) The direction of the vector is an angle that
is measured relative to north, south, east, or west.
N of W
N of E
S of W
S of E
 W of S
S
1
E of N
E of S
E
Addition of Vectors
Vector addition is used to determine the result of 2 or more vectors of the same
kind acting on an object. To do this, join the individual vectors head to tail
without changing their length or direction. The resultant (R) is drawn from the
“open” tail to the “open” head such that its head touches a head!!!
Examples of Vector Addition
+
+
=
+
=
R
R
Graphical Method of Addition
Adding vectors graphically requires the use of a scale, a protractor, and a ruler.
The scale is used to represent the magnitude (let 1 cm = _______ ). The protractor
is needed to measure the direction, and the ruler is needed to measure the
magnitude. Some examples of graphical vector addition are shown below.
Graphical Vector Addition Examples
For each of the following problems, find the resultant. Be sure to state its magnitude
(using the scale provided & a ruler) and direction (using a protractor) relative to
north, south, east, or west.
1.
1 cm = 10 meters
+
2
(~ 45m, 36 N of W)
3
2.
1 cm = 5 N
+
(~ 15 N, 41 S of W)
3.
1 cm = 8 m/s
+
+
(~ 36 m/s, 34 S of E)
4.
1 cm = 20 lb
+
+
(~100 lb, 13 N of E)
4
Vector Components: Once we have the resultant of a vector, we can further analyze
it by determining its components. The components of a vector represent how much
of the vector “acts” horizontally (vx) and how much of it acts vertically (vy). Visually,
the components are the legs of a right triangle in which the resultant is the
hypotenuse. The direction of the components is simply the horizontal and vertical
directions already expressed in the resultant!!!
5
Illustration of Vector Components :
R
R
vy
vy
vx
vx
Vector Components Examples
For each of the following, (1) draw the resultant, R. (You select a scale!!!)
(2) determine the components, vx and vy of R.
R = 35 m, 24 S of E
(32 m, east; 14 m, south)
R = 42 m/s, 30 N of E (36 m/s, east
21 m/s, north)
R = 15 N, 42 S of W
R = 27 lb, 63 N of W
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(11 N, west; 10 N, south)
(24 lb, north; 12 lb, west)
Graphical Vector Physics Problems
For each of the following, (1) choose a scale (let 1 cm = ______ )
(2) find the resultant, R
(3) find the components, vx and vy
1.
Kim flies east at 100 mph while a wind blows at 30 mph, 45 S of E .
(123 mph, 10° S of E)
(121 mph E, 21 mph S)
2.
Hannah sails her boat east across a river at 8 mph. The current is moving
north (upstream) at 12 mph and a breeze is blowing 25° N of E at 15 mph.
(28 mph, 41° N of E)
( 21mph E, 18 mph,
N)
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3.
John drives 8 miles at 37° N of W, and then 6 miles in a direction 20° E of N.
(11 miles, 18° W of N)
( 10 mi N ; 3.5 mi W)
4.
Kim is back in her plane again and flies 45° N of W at a speed of 500 mph. A
wind is gusting at 60 mph in a direction 30° N of E.
(505 mph, 53° N of W)
( 300 mph W ; 400 mph N)
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5.
Steph and Louisa build a hot air balloon in their backyard. They launch it
and head downwind (south) at 10 m/s. Matt steals Kim’s plane and flies by
them causing a second gust of wind blowing at 13 m/s at 56° E of S.
(20 m/s, 30° E of S)
(10 m/s E; 17.3 m/s S)
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Trigonometry Method of Vector Addition
A simpler way to add vectors in order to find the resultant R and its
components vx and vy is to use trigonometry. The use of trig means that you
don’t need a scale, don’t need a ruler, and don’t need a protractor! All you need
is your calculator in order to evaluate sine and cosine.
Here’s what to do:

Make and label a sketch of each vector.
If there are 2 vectors such that they and their resultant form a right triangle, then use
Pythagoras and Trig to find magnitudes and direction. Voila!
If the 2 vectors and their resultant don’t form a right triangle or there are more than
2 vectors, then:

For each vector, find the horizontal component, vx, by using cos 
where  is the angle made with the horizontal (or x-axis).
Be careful to call right (+) and left (-)!!!

Repeat the above step to find the vy for each vector using sin 
Be careful to call up (+) and down (-)!!!

Add all of the individual vx’s to get the resultant vx.

Add all of the individual vy’s to get the resultant vy.

Use Pythagoras applied to the resultant components to find the magnitude of R.

Use SOH CAH TOA to find the direction of R.
Special case for force vectors
When an object is in equilibrium, the force vectors acting on that object add up to
zero ( F = 0).
Visually, this means that all of the force vectors meet head to tail…there is no
room to draw a resultant!
Mathematically, this means that both the x components and the y components
add up to zero!
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Physics Trig Problems
1.
Brittany’s horse runs at 20 mph in a direction 58° S of E. Find its components.
(vx = 10.5 mph;
vy = 17 mph)
2.
Mike is bored and wants to go to the beach. He borrows Kim’s plane and tries
to fly east. But a 50 mph southerly wind knocks the plane off course by 17°.
Find the plane’s resultant speed and its original speed.
(170 mph, 163 mph)
3.
John’s car stalls. Find the components of a push applied to it the push is 125 lb
in a direction 30° N of E.
(vx = 108 lb;
vy = 62 lb)
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4.
Suppose the displacement between your home and WHHS is 4 miles in a
direction 48° N of W. What are the components of your displacement from
home when you are at school?
(vx = 2.7 mi ;
vy = 3 mi)
5.
Pete needs the plane to go skiing in Utah. He flies in a southwesterly direction
and has a southern component of speed of 200 mph and a resultant speed of
350 mph. Find his western component of speed and exact direction.
(vx = 287 mph
55° W of S)
6.
A boat tries to cross a stream at 10 m/s. The current downstream is 3 m/s. Find
the resultant speed and direction of the boat.
3 m/s
10 m/s
(10.4 m/s , 17 S of E)
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7.
A 60 N force is directed at 30 N of E. A 30 N force is directed at 45 N of E.
Find the resultant force.
(89 N, 35 N of E)
8.
Andy and Joe go hiking at the Water Gap. Starting at the trailhead, they walk
10 miles at 53 S of W, then 7 miles at 20 N of W. Find their displacement from
the trailhead.
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(13.8 mi, 24 S of W)
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9.
Kerri and Tara come to John’s aid. One of them pushes his car with 400 N at
30 N of W. Another pushes west with 340 N. The third pushes with 520 N at
60 S of W. Find their resultant force on the car.
(978 N, 15 S of W)
10.
Hannah’s boat has a top speed of 5 m/s in still water. She is sailing east
across a river. The current of the river moves south at 3 m/s. Meanwhile a
breeze is blowing at 4 m/s in a direction of 30 N of E. Find the boat’s
resultant motion.
(8.5 m/s, 7 S of E)
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11.
Three forces act on an object to produce equilibrium, which means that the
vector sum of the forces = 0. If one force is 33 N, north and the second force is
44 N at 60 N of E, find the 3rd force needed to produce equilibrium.
(note – this 3rd force is known as the equilibrant)
(74 N, 73 S of W)
12. A force of 20 N west and then 30 N at 53 N of W is applied to a crate. Find the
3rd force vector needed to keep the crate at rest.
(45 mi, 32 S of E)
13. 3 people push on a crate in such a way as to keep the crate in equilibrium. One
person pushes with a force of 50 N at 40 S of E. Another person pushes with a
force of 80 N at 60 N of E. Find the force exerted by the 3rd person.
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(86 N, 25 S of W)
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