Vectors

1
Vectors

• An essential component of your physics toolbox.

2
Vectors vs. Scalars

• Vector physical quantity that has both magnitude
  and direction
• Examples displacement, velocity, acceleration
• Scalar physical quantity that has magnitude but
  no direction
• Examples distance, speed, time, mass

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Vectors vs. Scalars
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Representing Vectors

• Symbols in text
• boldface
• Symbols on paper/board
• arrow above symbol
• Graphically
• draw an arrow
• length represents magnitude
• angle and head of arrow shows direction (angle)
• label vector with quantity and/or value

v1 3 m/s
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Adding and Subtracting Vectors

• Only vectors of the same quantity and units may
  be added or subtracted
• You cant add a velocity vector and a
  displacement vector
• You cant add a velocity in m/s with a velocity
  in mph
• Resultant vector that represents the sum of two
  or more vectors

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Graphical Addition of Vectors

• Draw all vectors to scale.
• Add vectors using Head-To-Tail Method
• Draw 1st vector
• Begin tail of 2nd vector at head of 1st vector
• Lather, rinse, repeat
• Draw resultant vector from tail of 1st vector to
  head of last vector. Label resultant.

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Resultant of Collinear Vectors
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Resultant of Perpendicular Vectors
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Properties of Vector Addition

• Vectors can be moved parallel to themselves in a
  diagram.
• Useful for switching from Head-To-Tail Method to
  Parallelogram Method
• Vectors can be added in any order
• Resultant will always be the same.
• To subtract a vector, add its opposite.
• Multiplying or dividing vectors by scalars
  results in vectors.
• velocity (vector) displacement (vector) / time
  (scalar)
• accel (vector) change in velocity (vector) /
  time (scalar)

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Vectors Can Be Added In Any Order
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Vectors Can Be Added In Any Order
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Vectors Can Be Added In Any Order
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Vector ComponentsGraphical Representation

• projection of vector along x y axes
• how much in the x direction, how much in the y
  direction
• drop projection lines to x y axes
• draw components along axes

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Resultant of Two Non-90? Vectors

• USE THE COMPONENT METHOD
• Add x components
• Add y components

15
Trigonometry Primer

• Trigonometric functions relate measurements in
  right triangles.
• Much of our work with vectors can be visualized
  with right triangles, so we need to know how to
  do trigonometry.

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Trig Primer Pythagorean Thm.

• The Pythagorean Theorem relates the lengths of
  the sides of a right triangle.

c
b
a
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Trig Primer SOH CAH TOA
hyp
opp
?
adj
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Vector ResolutionFinding Vector Components

• Trigonometric Method
• Visualize vector and its components as a right
  triangle
• Apply Pythagorean Theorem SOHCAHTOA to find
  components (missing sides)

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Vector AdditionFinding a Resultant Vector

• Magnitude
• Find the sum of all x-components and all
  y-components
• Use Pythagorean Theorem to find resultant from
  components.

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Vector AdditionFinding a Resultant Vector

• Direction
• Apply inverse trig functions to find angle