Resolving Vectors Mathematically

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Resolving Vectors Mathematically
Verna RichardCPHS Physics 2007
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Mathematical Resolution

• Vectors can be resolved mathematically to the
  same level of accuracy that they can by using a
  scaled drawing.

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Mathematical Resolution

• We use trigonometric functions to determine
  magnitude (amount of motion, force, etc.) and
  direction (angle of motion, force, etc.)

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Mathematical Resolution

• To determine magnitudes of resultant vectors in
  right triangles, it is easiest to use the
  Pythagorean Theorem.
• c2 a2 b2 where a and b are component vectors
  (legs of the triangle) and c is the resultant
  vector (hypotenuse of the triangle).
• Remember that UPPERCASE letters indicate ANGLES
  while lower case letters indicate lengths of the
  sides.

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Mathematical Resolution

• For special right triangles (45 - 45 - 90 and
  30 - 60 - 90), you may use their rules for
  determining magnitudes of sides, as well.
• 30-60-90 45-45-90

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Mathematical Resolution

• For non-right triangles, it is best to use the
  Law of Sines or the Law of Cosines.
• Law of Sines
• sin A sin B sin C
• a b c
• You may compare A to B, B to C, or A to C, but
  you do not need all three in the comparison. Set
  three knowns up as a proportion and solve for the
  unknown.

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Mathematical Resolution

• Law of Cosines
• c2 a2 b2 2ab(cos C)
• a2 b2 c2 2bc(cos A)
• b2 a2 c2 2ac(cos B)
• cos A b2 c2 a2
• 2bc
• cos B a2 c2 b2
• 2ac
• cos C a2 b2 c2
• 2ab

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Mathematical Resolution

• The best way to find angles of resultant vectors
  for right triangles is by using the formulas for
  sines, cosines, and tangents and then converting
  back to the actual angle using the inverse
  functions of each.
• sin T opposite leg/hypotenuse
• cos T adjacent leg/hypotenuse
• tan T opposite leg/adjacent leg
• Oscar had a hunk of apples.

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Mathematical Resolution

• Example Problem 1
• T-Boy rode his bike 18.2 km due south and turned
  due east and rode to Clotilles house, which was
  another 7.6 km away. What was the magnitude and
  direction of his displacement?

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Example Problem 1

• First, draw a diagram displaying the motion.

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Example Problem 1

• Then find the magnitude of the displacement using
  the Pythagorean Theorem.
• c2 a2 b2
• c2 (18.2 km)2 (7.6 km)2
• c2 331.24 km2 57.76 km2
• c2 389 km2
• c 19.72 km 20. km 2.0 X 10 km

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Example Problem 1

• Then use the information given in the problem to
  solve the angle. We were given sides a and b,
  which are the sides adjacent to and opposite of
  the directional angle, so we will use the tangent
  formula to find the angle.
• tan T opposite/adjacent
• tan T 7.6 km/18.2 km
• tan T 0.4176 SO
• tan-1 T 22.7 23

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Answer to Example Problem 1

• We know that
• The magnitude of displacement is 20. km
• The angle of displacement is 23
• Movement is more south than west
• Final answer should be represented as 20. km, 23
  E of S.