Adding Vectors by the
Component Method
Feel free to use to accompanying
notes sheet.
Adding Vectors by the Component Method
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Yesterday we added vectors which
were at right angles to one another.
What would happen if the vectors
were not at right angles?
A crow (who apparently isn’t aware
that he should only fly in straight
lines) first flies 10 km at 60° N of E.
The he flies 25 km at 10° N of E.
What is his total displacement?
Adding Vectors by the Component Method –
The Strategy
B = 25 km @ 10° N of E
10°
A = 10 km @ 60° N of E
60°
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Obviously the answer is not 35 km,
so what must be done?
Adding Vectors by the Component Method
B = 25 km @ 10° N of E
R = resultant
A = 10 km @ 60° N of E
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Obviously the answer is not 35 km,
so what must be done?
Find the RESULTANT
Adding Vectors by the Component Method
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The vectors need to be added
vectorally.
Both vectors need to be RESOLVED
into their components.
The components are then added
together to find the resultant.
A to be resolved into Ax and Ay.
B to be resolved into Bx and By.
Adding Vectors by the Component Method –
Resolve each vector into components
B = 25 km @ 10° N of E
A = 10 km @ 60° N of E
Adding Vectors by the Component Method –
Resolve each vector into components
B = 25 km @ 10° N of E
By
Bx
Ay
A = 10 km @ 60° N of E
Ax
Adding Vectors by the Component Method
B = 25 km @ 10° N of E
By
Bx
Ay
A = 10 km @ 60° N of E
Ax
Ax = 10cos60°
Bx = 25cos10°
Ay = 10sin60°
By = 25sin10°
Adding Vectors by the Component Method
Bx
By
Ay
Ax
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The x-components and y-components can each be
considered legs of the resulting triangle.
Adding Vectors by the Component Method
By
Ay
Bx
Ax
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The x-components and y-components can each be
considered legs of the resulting triangle.
Adding Vectors by the Component Method –
Construct the “resulting triangle” from components
By
Ry
R
Ay
Rx
Bx
Ax
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Rx = Ax + Bx
Ry = Ay + By
Adding Vectors by the Component Method
Rx = 29.60
Ry = 13.00
By R = ??
Ry
R
Ay
Rx
Bx
Ax
How do we find R?
YES!! Pythagorean Theorem: R2 = Rx2 + Ry2
Adding Vectors by the Component Method
By
Ry
R
Ay
Rx
Ax
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R2 = RX2 + RY2
Bx
Adding Vectors by the Component Method
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Math – IN DEGREE MODE
Ax = 10cos60° = 5.00 km
Ay = 10sin60° = 8.66 km
Bx = 25cos10° = 24.6 km
By = 25sin10° = 4.34 km
Rx = Ax + Bx = 29.6 km
Ry = Ay + By = 13.00 km
R = 32.33 km
Adding Vectors by the Component Method
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Another way to organize data
x-axis
y-axis
A
10cos60°
10sin60°
B
25cos10°
25sin10°
R
Adding Vectors by the Component Method
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Another way to organize data
x-axis
y-axis
A
5.00
8.66
B
24.62
4.34
R
Adding Vectors by the Component Method
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Another way to organize data
x-axis
y-axis
A
5.00
8.66
B
24.62
4.34
R
29.62
13.00
Adding Vectors by the Component Method
Rx = 29.60
Ry = 13.00
By R = 32.33
Ry
R
Ay
Rx
Bx
Ax
Is this it? 32.33 km?
Adding Vectors by the Component Method
Rx = 29.60
Ry = 13.00
By R = 32.33
Ry
R
Ay
Rx
Bx
Ax
Is this it? No!! Now we have to find the
direction.
Adding Vectors by the Component Method
Rx = 29.60
Ry = 13.00
R = 32.33
R
θ
Rx
tan θ = Ry/Rx
θ = tan-1(Ry/Rx)
Ry
Adding Vectors by the Component Method
Rx = 29.60
Ry = 13.00
R = 32.33
R
θ
Ry
Rx
θ = 23.71° N of E
How do we know it is North of East?
Adding Vectors by the Component Method
N (north)
A
W (west)
B
40° N of E
C
Which angle is 10° North of West? A, B or C?
Which angle is 10° West of North? A, B or C?
What is angle C?
S (South)
E (east)
Adding Vectors by the Component Method
B = 25 km @ 10° N of E
R
A = 10 km @ 60° N of E
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Final answer: 32.33 km @23.71° N of E
Adding Vectors by the Component Method
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Some helpful hints!
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Never use the original values after the
vector has been resolved.
Assign negative values to S and W
components of vectors (assuming N and E
are positive)
Always make sure you are in degree
mode.
Adding Vectors by the Component Method
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Some helpful hints!
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Always make sure you are in degree
mode.
Make sure you draw your vectors in the
correct directions initially.
To help, redraw a coordinate system at
the end of each vector.
Be organized, stay organized, & finish the
entire problem.
Adding Vectors by the Component Method
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Other examples
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A duck flies 10 m/s @ 30° S of W with a
wind blowing 5 m/s N. What is the
resulting velocity of the duck?
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Answer:
Adding Vectors by the Component Method
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Other examples
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A duck flies 10 m/s @ 30° S of W with a
wind blowing 5 m/s N. What is the
resulting velocity of the duck?
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Answer:
B
0 x & +y
A
-x & -y
Adding Vectors by the Component Method
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Other examples

A duck flies 10 m/s @ 30° S of W with a
R = ??
wind blowing 5 m/s N.
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B
Answer:
A
x-axis
y-axis
A
-10cos30
-10sin30
B
0
+5
R
Adding Vectors by the Component Method
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Other examples

A duck flies 10 m/s @ 30° S of W with a
R = ??
wind blowing 5 m/s N.
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B
Answer:
A
x-axis
y-axis
A
-8.66
-5
B
0
+5
R
Adding Vectors by the Component Method
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Other examples

A duck flies 10 m/s @ 30° S of W with a
R = ??
wind blowing 5 m/s N.
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B
Answer:
A
x-axis
y-axis
A
-8.66
-5
B
0
+5
R
-8.66
0
Adding Vectors by the Component Method
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Other examples

A duck flies 10 m/s @ 30° S of W with a
wind blowing 5 m/s N.
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Answer:
R = -8.66 m/s or 8.66 m/s W
B
R
A
x-axis
y-axis
-8.66
0
Adding Vectors by the Component Method
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ANY and ALL vectors can (and will)
be analyzed this way.
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Displacements
Velocities
Accelerations
Forces
Momentum
Adding Vectors by the Component Method
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Any questions?
These notes will be online
You MUST be good at vectors to
succeed / pass this class.
Ask questions (in class) whenever
necessary.