Representing Vector Quantities

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Chapter 4
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RepresentingVector Quantities

• Graphical representation
• Using an arrow to show both the length and
  direction
• Must have set scale
• Algebraic representation
• Using math to break down the aspects of the vector

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• Two displacements are equal when the two
  distances and directions are the same
• Resultant vector
• Equal to the sum of two or more vectors
• The answer or result from putting two or more
  vectors together

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Vectors

• Every vector physical quantity can be broken down
  into 2 distinct parts
• These parts are called components
• Each component can be treated as a vector
• The of parts is a function of the space that
  the given vector is within
• dimensions components
• Can be moved around in space as long as you dont
  change or disturb their magnitudes or directions

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Graphically Speaking
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Graphically Adding Vectors

• Place the tail of the 1st vector at the origin
• Determine the correct direction (angle)
• Draw the magnitude of the 1st vector in the
  appropriate direction
• Remember to use a set scale for the magnitude
• Put an arrow on the head of the vector
• At head of 1st arrow, draw a NEW x-y origin
• Follow same rules as above for graphing second
  vector
• Continue process until all vectors have been added

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GAV continued

• Once all vectors have been added, draw the
  resultant
• This is a dotted line from the original origin of
  the 1st vector to the final arrow (head) of the
  last vector added.
• Measure the angle of the resultant vector
• Measure the length of the resultant vector (head
  to tail) and convert using the scale used for the
  entire problem
• You should then have both a direction and a
  magnitude for you answer

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Analytically Speaking
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Equations to Remember

• sin T opp cos T adj
• hyp hyp
• tan T sin T opp
• cos T adj

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Equations to Remember

• Pythagorean Theorem
• A2 B2 C2
• Law of Cosines
• A2 B2 AB cos T C2

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• Every vector can be resolved into two components
• Vector resolution
• Process of breaking a vector into its components

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Vector A

• If we have vector A, we can break it down into 2
  parts.

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T

• 1 part in the x and 1 in the y
• Represented with Ax Ay

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T

• Look at this as a triangle. Where is your side
  A, your side B, your hypotenuse ?

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• Vector A is your hypotenuse
• Ax is your adjacent side
• Ay is your opposite side

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So in RESOLVING this vector

• sin T opp Ay
• hyp A
• cos T adj Ax
• hyp A
• tan T opp Ay
• adj Ax

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Generally Speaking

• Ay A (sin T)
• Ax A (cos T)
• T tan -1 (Ay)
• Ax

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  Vector A T Ax Ay
1 30 30  26 15
2 39 60  19.5 33.8  
3  39.6 45  28 28
4 13  22.6  12 5
5 48 72 14.8  45.7 
6 120 15  115.9 31.1
7 72 40  55.2 46.3
8  89.2 66.9  35 82
9  145.8 31  125 75
10 57.7  30 50 28.9 
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If adding several vectors

• Resolve each vector involved into their
  individual x and y components
• Get Sum of all x components
• Get Sum of all y components
• THESE (Sx and Sy) are the x and y components of
  the resultant

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Vector Resolving Table
Vector Mag. Dir. Fx Fy
(A, T) A T A (cos T) A (sin T)
A 115 30 100 58
B 90 70 31 85




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Completing the Vector Resolving Table

• For each vector, calculate the Fx Fy
• To find the RESULTANT
• Sum the Fx Fy
• Remember a2 b2 c2
• The Resultant Magnitude (R)
• R2 SFx2 SFy2
• The Resultant Angle (T)
• Tan T SFy
• SFx

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Vector Resolving Table
Vector Mag. Dir. Fx Fy
(A, T) A T A (cos T) A (sin T)




total total total SFx SFx
Resultant (R, T) Resultant (R, T) Resultant (R, T)