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Lecture 03 Vectors
 

Lecture 03 Vectors

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  • R = (32.6 m, 143.0o)1BSN1
  • 5.31 km, S5.31 km, N

Lecture 03 Vectors Lecture 03 Vectors Presentation Transcript

  • Vectors
  • Surveyors use accurate measures of magnitudes and directions to create scaled maps of large regions.
    Vectors
  • Identifying Direction
    N
    50o
    60o
    E
    W
    60o
    60o
    S
    A common way of identifying direction is by reference to East, North, West, and South. (Locate points below.)
    Length = 40 m
    40 m, 50o N of E
    40 m, 60o N of W
    40 m, 60o W of S
    40 m, 60o S of E
  • Identifying Direction
    N
    45o
    N
    E
    W
    E
    W
    50o
    S
    S
    Write the angles shown below by using references to east, south, west, north.
    500 S of E
    450 W of N
  • 90o
    90o
    R
    q
    50o
    180o
    180o
    0o
    0o
    270o
    270o
    Vectors and Polar Coordinates
    Polar coordinates (R,q) are an excellent way to express vectors. Consider the vector 40 m, 500 N of E, for example.
    40 m
    R is the magnitude and q is the direction.
  • 90o
    0o
    180o
    50o
    60o
    60o
    60o
    3000
    210o
    270o
    120o
    Vectors and Polar Coordinates
    Polar coordinates (R,q) are given for each of four possible quadrants:
    = 40 m, 50o
    = 40 m, 120o
    = 40 m, 210o
    = 40 m, 300o
  • y
    (-2, +3)
    (+3, +2)
    +
    +
    x
    -
    Right, up = (+,+)
    Left, down = (-,-)
    (x,y) = (?, ?)
    -
    (+4, -3)
    (-1, -3)
    Rectangular Coordinates
    Reference is made to x and y axes, with + and -numbers to indicate position in space.
  • R
    y
    q
    x
    Trigonometry Review
    Application of Trigonometry to Vectors
    Trigonometry
    y = R sin q
    x = R cos q
    R2 = x2 + y2
  • q
    Finding Components of Vectors
    A component is the effect of a vector along other directions. The x and y components of the vector are illustrated below.
    = A cosq
    = A sin q
    Finding components:
    Polar to Rectangular Conversions
  • Example 2:A person walks 400.0 m in a direction of 30.0oSof W (210o). How far is the displacement westand how far south?
    30o
    400 m
    The x-component (W) is adjacent:
    = -A cosq
    The y-component (S) is opposite:
    = -A sinq
  • Vector Addition
    Resultant ( )
    - sum of two or more vectors.
    Vector Resolutions:
    01. GRAPHICAL SOLUTION
    – use ruler and protractor to draw and measure the scaled magnitude and angle (direction), respectively.
    02. ANALYTICAL SOLUTION
    - use trigonometry
  • q
    Example 11:A bike travels 20 m, E then 40 m at 60o N of W, and finally 30 m at 210o. What is the resultant displacement graphically?
    Graphically, we use ruler and protractor to draw components, then measure the Resultant R,q
    C = 30 m
    B = 40 m
    30o
    R
    60o
    f
    A = 20 m, E
    R = (32.6 m, 143.0o)
    Let 1 cm = 10 m
  • Cy
    30o
    R
    Ry
    60o
    f
    0
    q
    Ax
    Rx
    Bx
    Cx
    A Graphical Understanding of the Components and of the Resultant is given below:
    Note: Rx = Ax + Bx + Cx
    By
    B
    Ry = Ay + By + Cy
    C
    A
  • Resultant of Perpendicular Vectors
    Finding resultant of two perpendicular vectors is like changing from rectangular to polar coord.
    R
    y
    q
    x
    R is always positive; q is from + x axis
  • First Consider A + B Graphically:
    B
    B
    R
    B
    A
    A
    Vector Difference
    For vectors, signs are indicators of direction. Thus, when a vector is subtracted, the sign (direction) must be changed before adding.
    R = A + B
  • Now A – B: First change sign (direction) of B, then add the negative vector.
    B
    B
    -B
    A
    R’
    -B
    A
    A
    Vector Difference
    For vectors, signs are indicators of direction. Thus, when a vector is subtracted, the sign (direction) must be changed before adding.
  • Subtraction results in a significant difference both in the magnitude and the direction of the resultant vector. |(A – B)| = |A| - |B|
    Comparison of addition and subtraction of B
    B
    B
    A
    R
    R’
    -B
    B
    A
    A
    Addition and Subtraction
    R = A + B
    R’ = A - B
  • A – B; B - A
    +A
    -A
    +B
    -B
    A 2.43 N
    B 7.74 N
    Example 13.Given A = 2.4 km, N and B = 7.8 km, N: find A – B and B – A.
    A - B
    B - A
    R
    R
    (2.43 N – 7.74 S)
    (7.74 N – 2.43 S)
    5.31 km, S
    5.31 km, N