4-2 Components of Vectors

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4-2 Components of Vectors
Choosing a Coordinate System
No correct answer on how to choose
 Choosing a Correct coordinate system will better
allow certain problems to be solved
 Similar to laying a grid down on a sheet of paper
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◦ Choose the center of the grid (origin)
◦ On Earth: generally us the four Directions
◦ Through air: x-axis chosen to be horizontal and y-axis
chosen to be vertical
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Direction of a vector is defined as the angle made
with the x-axis measured counterclockwise
Components
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A vector may be broken up into two
component vectors.
◦ Parallel with x-axis (Ax)
◦ Parallel with y-axis (Ay)
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Original Vector A is the sum of the two
component vectors
◦ A = Ax + Ay
◦ Vector Resolution is the process of breaking a
vector into it’s components
Components
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All algebraic calculations involve only the
components of vectors, not the vector itself
◦ Find by using Trigonometry
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Components are calculated where angle θ is
measured counterclockwise from positive x-axis
◦ Ax = A cos θ; cos θ = (adjacent side/hypotenuse) = Ax / A
◦ Ay = A sin θ; sin θ = (opposite side/hypotenuse) = Ay / A
Practice
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A bus travels 23.0 km on a straight road that is 30o N of
E. Where are the east and north components of its
displacement?
Sketch problem
Known: A = 23.0 km
= 30o
Unknown: Ax = ?
Ay = ?
Calculation
◦ Ax = A cos θ;
Ax = (23.0 km)cos 30o
◦ = +19.0 km
◦ Ay = A sin θ;
Ay = (23.0 km)sin 30o
◦ = +11.5 km
Practice
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What are the components of a vector of a magnitude
1.5 m at an angle of 35o form the positive x-axis?
Algebraic Addition of Vectors
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Two or more vectors (A, B, C,…) may be added by first
resolving each vector to its x & y components
◦ The x-components are added to form the resultant (Rx)
◦ Rx = Ax + Bx + Cx + …
◦ The y-components are added to form the resultant (Ry)
◦ Ry = Ay + By + Cy + …
◦ To find the angle or direction of the resultant, the
tangent of the angle that the vector makes with the xaxis is the following (tan θ = Ry/Rx)
◦ Find the angle by using the tan-1 key
Algebraic Addition of Vectors
Cy
C
R
By
Ay
A
Ax
B
Bx
Cx
Rx
Ry
Practice
A GPS reciever told you that your home was 15.0 km at
a direction 40o North of West, but the only path led
directly north. If you took that path and walked 10.0
km, how far, and in what direction would you have to
walk to reach your home?
 Draw the resultant vector, R from your original location
to home
 Draw A, the known displacement
 Draw B, the unknown displacement
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A GPS reciever told you that your home was 15.0 km at a direction
40o North of West, but the only path led directly north. If you took
that path and walked 10.0 km, how far, and in what direction would
you have to walk to reach your home?
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Rx = R cos θ ; (15.0 km) cos θ 140o;
= -11.5 km
Ry = R sin θ ; (15.0 km) cos θ 140o;
= +9.6 km
Ax = 0.0 km; Ay = 10.0 km
R = A + B; so B = R – A
Bx = Rx – Ax = (-11.5 km) – (0.0 km) = -11.5 km
By = Ry – Ay = (9.6 km) – (10.0 km) = -0.4 km
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B = √Bx2 + By2 = √(-11.5 km)2 + (-0.4 km)2
= 11.5 km
tan θ = By/Bx = (-0.4 km)/(-11.5 km) = +0.035
θ = tan-1(+0.035) = 2.0o
B = 11.5 km, 2.0o south of west

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