VECTORS

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VECTORS
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Vectors

• A person walks 5 meters South, then 6 meters
  West.
• How far did he walk?

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Vectors

• A person walks 5 meters North, then 6 meters
  East.
• How far did he walk?
• What is his displacement?

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Scalars Vectors

1. A scalar is a quantity that has magnitude (size)
   only.
2. Vectors are quantities that have size and
   direction.

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Examples

• Scalars 5 m. 7 m/s.
• Vectors 5 m South. 7 m/s up.

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Vectors

• Vectors have magnitude5 m 7 m/s
• AND
• Vectors have directionSouth Up

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Drawing vectors.

• Vectors are drawn as arrows.
• Length of the arrow is proportional to the
  magnitude of the vector.
• (The bigger the number,
• the longer the arrow!)

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Vector Arithmetic

• Vectors can be multiplied by numbers.
• If A 5 m North, 2A 10 m North

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Vector Arithmetic

• Negative signs flip the direction.

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Vectors

• Vectors can be added together.
• To show vector addition graphically, you need to
  draw the vectors
• To scale
• Tip to Tail.
• The red line is the sum of the other
  two vectors and is called the resultant.
• Adding vectors gives a displacement, not a
  distance.

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Graphic Addition Examples
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• Switch to Notebook to draw vector answers or
  write on White Board on top of projection.

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Vector Arithmetic
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Common notation

• A vector will usually be given as a length at
  some angle.
• I.e. 4 m _at_ 60? or 7 m _at_ 235?

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Common Notation

• The angle listed will be measured counter
  clockwise from the x axis.(unless otherwise
  specified)

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Common Notation

• The angle listed will be measured counter
  clockwise from the x axis.(unless otherwise
  specified)

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ADDING VECTORS ALGEBRAICALLY

1. Adding vectors using algebra gives us a much more
   exact number than doing it graphically.
2. To add vectors algebraically, you need to
   remember a little geometry and trigonometry.

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PERPENDICULAR VECTORS

• When vectors are at right angles to each other,
  we can find both the size and direction.
• We find the magnitude of the resultant by using
  the Pythagorean Theorem (a2 b2 c2 ).
• We find the direction (angle) of the resultant by
  using inverse cosine or inverse tangent functions.

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Find the resultant.

• A 6 m _at_ 0 deg.
• B 9 m _at_ 90 deg.
• What is A B?

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Adding parts.

• Add parts together to get same answer as adding
  wholes together, just like adding vectors. For
  example
• Adding Wholes 2 7 9
• Adding Parts 2 1 1 and 7 3 4 so
• (1 1) (3 4) 9 and is the same as
  adding wholes above.

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Components

• Any vector can be treated like the sum of 2
  vectors or broken into 2 parts.
• A vector can be treated like a horizontal (x)
  vector and a
• vertical (y) vector added together.
• X means a unit vector in the X direction.
• y means a unit vector in the y direction.

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Components

• Vector A
• Ax is the horizontal component of A
• Ay is the vertical component of A

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Trig Review

• Vector A of length A is the hypotenuse.
• X is the length or magnitude of the
• X component (vector Ax).
• Y is the length or magnitude of the
• Y component (vector Ay).
• Cos Q (X/A) Sin Q (Y/A) or
• X A Cos Q Y A Sin Q
• Inverse Cos is written as Cos -1 .
• Inverse Cos asks what angle produces that Cos.
• Cos -1 (X/A) gives Q or Tan -1 (Y/X) gives Q

y
Q
X
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ADDING NON-PERPENDICULAR VECTORS

• When vectors are applied at different angles, we
  need to break them into their vertical (y) and
  horizontal (x) pieces.
• Then we can find (S is sigma sum)
• S x and S y
• Use Pythagorean Theorem to find resultant
  magnitude (M).
• (S x)2 (S y)2 M2
• Use Inv Cos S x to find resultant
• M direction angle.

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EXAMPLE

• A 50 km at 110 deg
• B 30 km at 60 deg

A 50 km
B 30 km
? 110o
? 60o
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Example

• We want A B

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EXAMPLE

• The resultant is the vector made from the
  combined components of the two original vectors.

A 50 km
B 30 km
? 110o
? 60o
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• X Ax BX The Horizontal of A The
  horizontal of B
• X 50cos(110) 30 cos(60)
• X -17.1 15 -2.1

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• Y AY BY The Vertical of A The Vertical of
  B
• Y 50sin(110) 30 sin(60)
• Y 47.0 26.0 73.0

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• C -2.1 Horizontal
• 73 Vertical

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How long is C?

• C2 (-2.1)2 (73)2
• C 73.03

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What is the angle of C?

• cos Q -2.1 / 73.03
• Q cos -1(-2.1 / 73.03)
• Q 91.6 degrees

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Find the resultant

• C 5 km at 30 deg.
• D 9 km at 45 deg.
• Find C D