Using Polar Plot Graph Paper to
Resolve Vector Addition
By Jack Graham
March 2013
Key Idea to Learning Graphic Vector Addition:
1. All vectors acting on the same point can be represented
by an “x-component” and a “y-component”.
“y-component”
Y
“x-component”
X
Special Vector Analysis Graph Paper
This is a polar plot graph.
Each Ring is 1.0 Newton or
the weight of a 100g mass.
1 2 3 4 5 6 7 8 9 10
Suppose we want to add
two vectors, A at 0⁰ of 5 N
To B at 120⁰ of 7 N
We can draw the two vectors starting at the origin and
extending the arrow head out to the 5 N ring and 7 N ring.
Special Vector Analysis Graph Paper
Y
Suppose we want to add
two vectors, A at 0⁰ of 5 N
To B at 120⁰ of 7 N
Adding a normal x-y axis
to it allows us to easily
find x & y component
vectors.
10
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-10 -9 -8 -7 -6 -5 -4 -3 -2 -1-1
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X
Special Vector Analysis Graph Paper
Y
Suppose we want to add
two vectors, A at 0⁰ of 5 N
3”x 5” Card
To B at 120⁰ of 7 N
Adding a normal x-y axis
to it allows us to easily
find x & y component
vectors.
Use the edge of a card to
drop a dashed line from
the head of each vector to
the X and Y axis.
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-10 -9 -8 -7 -6 -5 -4 -3 -2 -1-1
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X
Special Vector3”x
Analysis
Graph
Paper
5” Card
Y
Suppose we want to add
two vectors, A at 0⁰ of 5 N
To B at 120⁰ of 7 N
Adding a normal x-y axis
to it allows us to easily
find x & y component
vectors.
Use the edge of a card to
drop a dashed line from
the head of each vector to
the X and Y axis.
10
9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1-1
-2
-3
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-5
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-10
X
Special Vector Analysis Graph Paper
Y
Suppose we want to add
two vectors, A at 0⁰ of 5 N
To B at 120⁰ of 7 N
The only Y component is
drawn in from the origin
to the dashed line from B
The A vector has no Y
component.
The X component of the B
vector is – 3.5
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-10 -9 -8 -7 -6 -5 -4 -3 -2 -1-1
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The X component of the A
vector is the whole A vector
X
Special Vector Analysis Graph Paper
Y
Suppose we want to add
two vectors, A at 0⁰ of 5 N
To B at 120⁰ of 7 N
To find the X-component
of R the Resultant Vector
we combine both of the
X-components adding as
signed numbers:
+5 – 3.5 = +1.5
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We draw the Rx-component.
We recognize the By-component as the Ry-component
(changed to R color here)
X
Special Vector Analysis Graph Paper
3”x 5” Car
Y
10
Suppose we want to add two
9
8
vectors, A at 0⁰ of 5 N
7
6
To B at 120⁰ of 7 N
5
4
Using a 3” by 5” card as a
3
2
guide we draw
1
1
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-1
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perpendicular lines from the
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-3
ends of the R-components.
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X
Special Vector Analysis Graph Paper
Y
10
Suppose we want to add two
9
8
vectors, A at 0⁰ of 5 N
7
6
To B at 120⁰ of 7 N
5
4
Using a 3” by 5” card as a
3
2
guide we draw
1
1
-10
-9
-8
-7
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-5
-4
-3
-2
-1
-1
perpendicular lines from the
-2
-3
ends of the R-components.
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-10
3”x 5” C
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X
Special Vector Analysis Graph Paper
Y
10
Suppose we want to add
9
8
two vectors, A at 0⁰ of 5 N
7
6
To B at 120⁰ of 7 N
5
4
Using a 3” by 5” card as a
3
2
guide we draw
1
1
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
-1
perpendicular lines from the
-2
-3
ends of the R-components.
Where these perpendiculars
intersect is the head of the
resultant which we draw.
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We can see from the graph the R vector is 6.2 N at 76⁰.
X
Key Ideas to Learning Graphic Vector Addition:
1. All vectors acting on the same point can be represented
by an “x-component” and a “y-component”.
2. X-components to the right are positive numbers.
3. X-components to the left are negative numbers.
4. All the X-components can be algebraically added (like
signed numbers) to get the X-component of the resultant.
5. Y-components going up are positive numbers.
6. Y-components going down are negative numbers.
7. All the Y-components can be algebraically added (like
signed numbers) to get the Y-component of the resultant.
8. Resultant vector’s magnitude and direction angle can be
found graphically from its “x-component” and “ycomponent”.
Practice Vector Addition
• Distribute the “Polar Plot Graphs Vector
Addition Using Component Vectors Practice
Worksheet 1”
• Distribute 3”x5” cards
• Do the six problems. When a student finishes,
check their work. If it is correct have them help
others and check the next students finished.
Continue until all have completed the solutions.
• Collect the papers and put them in the wire
basket on Mr. Graham’s desk.
Finding an Equilibrant
Y
10
Suppose we want to add
9
8
two vectors, A at 0⁰ of 5 N
7
6
To B at 120⁰ of 7 N
5
4
Using a 3” by 5” card as a
3
2
guide we draw
1
1
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
-1
perpendicular lines from the
-2
-3
ends of the R-components.
Where these perpendiculars
intersect is the head of the
resultant which we draw.
2 3 4 5 6 7 8 9 10
-4
-5
-6
-7
-8
-9
-10
We can see from the graph the R vector is 6.2 N at 76⁰.
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Finding an Equilibrant
An equilibrant force is a force vector that balances a force
vector or group of force vectors causing all the forces acting
at a point to sum to zero giving no change in motion.
Y
Given forces A, B, and C,
how can we determine the
size and direction of their
equilibrant E?
R
B
A
C
By finding the sum of the
given forces R and
reversing its direction we
find E.
E
X
Finding an Equilibrant
Given Force A at 0⁰ 4 N, B at 90⁰ 4 N, and C at 180⁰ 4 N,
determine their equilibrant.
Hints:
Use the method for adding two or three vectors to find their
resultant using special polar plot graphing methods, to add
the given vectors.
Then reverse the resultants components
To find the equilibrium force vector.
Use the force table to check your answer.
Practice Finding an Equilibrant
Distribute the blank Polar Plot Graphs for Vector Addition Using Component Vectors
Find the equilibrant for each of the following problems. Check your answers using
the force table. The first students done set up the force table solution for all to see.
1. Given Force A at 0⁰ 4.5 N, B at 60⁰ 4.5 N, and C at 200⁰
4.5 N, determine their equilibrant.
2. Given Force A at 0⁰ 4.5 N, B at 110⁰ 4.5 N, and C at 220⁰
4.5 N, determine their equilibrant.
3. Given Force A at 0⁰ 4.5 N, B at 60⁰ 8.5 N, and C at 200⁰
4.5 N determine their equilibrant.
4. Given Force A at 0⁰ 4.5 N, B at 110⁰ 8.5 N, and C at 220⁰
4.5 N, determine their equilibrant.
5. Given Force A at 0⁰ 8.5 N, B at 60⁰ 8.5 N, and C at 200⁰
8.5 N, determine their equilibrant.
6. Given Force A at 0⁰ 8.5 N, B at 110⁰ 8.5 N, and C at 220⁰
8.5 N, determine their equilibrant.