Engineering Fundamentals

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Engineering Fundamentals

• Session 6 (1.5 hours)

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Scaler versus Vector

• Scaler (??) described by magnitude
• E.g. length, mass, time, speed, etc
• Vector(??) described by both magnitude and
  direction
• E.g. velocity, force, acceleration, etc

Quiz Temperature is a scaler/vector.
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Representing Vector

• Vector can be referred to as
• AB or a or a
• Two vectors are equal if they have the same
  magnitude and direction
• Magnitudes equal a c or a c
• Direction equal they are parallel and pointing
  to the same direction

How about these? Are they equal?
4
Opposite Vectors

• magnitudes are equal,
• parallel but opposite in sense
• These two vectors are not equal
• Actually, they have the relation
• b -a

5
Rectangular components of Vector

• A vector a can be resolved into two rectangular
  components or x and y components
• x-component ax
• y-component ay
• a ax, ay

y
x
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Addition of Vectors
V1 V2
V2
V1
V1
V1
V1 V2
V2
V2
Method 2
Method 1
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Subtraction of Vectors
-V2
V1
V1 - V2
V1
-V2
V2
8
Scaling of vectors (Multiply by a constant)
V1
V1
2V1
0.5V1
V1
-V1
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Class work

• Given the following vectors V1 and V2. Draw on
  the provided graph paper
• V1V2
• V1-V2
• 2V1

V1
V2
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Class Work

• For V1 given in the previous graph
• X-component is _______
• Y-component is _______
• Magnitude is _______
• Angle is _________

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Rectangular Form and Polar Form

• For the previous V1
• Rectangular Form (x, y)
• 4, 2
• Polar Form (r, ?)
• v20 26.57 ? or (v20 , 26.57 ? )

y-component
x-component
angle
magnitude
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Polar Form ? Rectangular Form

• Vx V cos ?
• Vy V sin ?

magnitude of vector V
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Example

• Find the x-y components of the following vectors
  A, B C
• Given
• A2, ?A 135o
• B4, ?B 30o
• C2, ?C 45o

A
?A
C
?B
?C
B
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Example (Contd)

• For vector A,
• Ax2 x sin(135o) ?2, Ay2 x cos(135o)-?2
• For vector B,
• Bx4 x sin(210o) -4 x sin(60o)-2,
• By4 x cos(210o) -4 x cos(60o)-2?3
• For vector C,
• Cx2 x sin(45o) ?2, Cy2 x cos(45o)-?2

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Example

• What are the rectangular coordinates of the point
  P with polar coordinates (8, p/6)
• Solution
• use xrsin ? and yrcos ?
• x8sin(p/6)8(?3/2)4?3
• y8cos(p/6)8(1/2)4
• Hence, the rectangular coordinates are (4?3,4)

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Rectangular Form -gt Polar Form

• Given (Vx, Vy), Find (r, ?)
• R ? Vx2 Vy2 (Pythagorus Theorm)
• ? tan -1 (Vx / Vy) ? Will only give answers in
  Quadrants I and VI
• Need to pay attention to what quadrant the vector
  is in

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How to Find Angle?

• Find the positive angle Ø tan-1 (Vy/Vx)
• ? Ø or 180-Ø or 180Ø or Ø, depending on what
  quadrant.

Absolute value (remove the negative if any)
180-Ø
Ø
Ø
Ø
Ø
Ø
-Ø
180Ø
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Classwork

• Find the polar coordinates for the following
  vectors in rectangular coordinates.
• V1 (1,1) r____ ?_______
• V2(-1,1) r____ ?_______
• V3(-1,-1) r____ ?_______
• V4(1,-1) r____ ?_______

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Class work

• a (6, -10) r____ ?_______
• b (-6, -10) r____ ?______
• c (-6, 10) r____ ?______
• d (6, 6) r____ ?_______

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Concept Map
V
notation
V
AB
Vectors
operations
Scalar multiplication
representation
Rectangular Form (Vx,Vy)
2 V
Subtraction-
conversion
V1 V2
Addition
Polar Form (r, ?)
V1 V2
Beware of the quadrant, and use of tan-1 !!!
Angle or phase
magnitude