Vectors

1
Vectors

• Chapter 3, Sections 1 and 2

2
Vectors and Scalars

• Measured quantities can be of two types
• Scalar quantities only require magnitude (and
  proper unit) for description. Examples distance,
  speed, mass, temperature, time
• Vector quantities require magnitude (with unit)
  and direction for complete description.
  Examples displacement, velocity, acceleration,
  force, momentum

3
Vector Addition

• When 2 or more vectors act on an object, the
  total effect is the vector sum
• Special math operations must be used with vectors
• Vector sum called the resultant
• Sum can be found using graphic methods (drawing
  to scale) or mathematical methods

4
Adding Vectors Graphically

• Choose a suitable scale for the drawing
• Use a ruler to draw scaled magnitude and a
  protractor for the direction
• Vectors can be moved in a diagram as long as
  their length and direction are not changed
• Vectors can be added in any order without
  changing the result

5
Adding Vectors Graphically

• Use head-to-tail method for series of sequential
  vectors where each successive vector begins where
  the preceding vector ended
• Also works for two vectors acting simultaneously
  at the same point, although drawing doesnt match
  the physical situation

6
Adding Vectors Graphically

• Resultant is a vector drawn from point of origin
  to tip of last vector
• Magnitude of resultant can be found by measuring
  and converting the measurement using the scale of
  the drawing
• Direction is found by measuring angle with a
  protractor

7
Adding Vectors Graphically

• Graphical method gives approximate values
  depending on drawing accuracy

One dimensional graphical addition of vectors
8
Subtracting Vectors

• To subtract a vector, add a negative vector, one
  having same magnitude but opposite direction

b
a b
a
-b
a
a
a - b
b
9
Parallelogram Method

• Vectors are drawn from a common origin
• Complete parallelogram by drawing opposite sides
  parallel to vectors
• Resultant is the diagonal of the parallelogram

10
Pros and Cons for Graphical Methods

• For simultaneous vectors like forces,
  parallelogram method gives a better picture of
  actual situation
• More difficult to draw accurately
• Better for sketches, not for measured drawings
• Head-to-tail method better for measuring

11
Parallelogram Method

• a b r

a
r
b
12
Adding Vectors Mathematically

• Exact values for vector sums using trig functions
  (tan mostly) and Pythagorean theorem
• Set up vectors on x-y coordinate system
• If vectors act at right angles, Pythagorean
  theorem gives resultant magnitude
• Direction can be found with tan-1 function

13
Resolving Vectors Into Components

• A vector acting at an angle to the coordinate
  axes can be resolved into x and y components that
  would add together to equal the original vector
• The x-component original magnitude times the
  cos of the angle measured from the x-axis
• The y-component original magnitude times the
  sin of the angle measured from the x-axis

14
Vector Components
vx (50 m/s)(cos 60o)
vy (50 m/s)(sin 60o)
15
Adding Non-perpendicular Vectors

• Resolve each vector into x and y components
• Add the x-components together and add the
  y-components together
• Use Pythagorean theorem and tan-1 function to
  find magnitude and direction of resultant

16
Adding Non-perpendicular Vectors
17
Adding Non-perpendicular Vectors
Rx 11.3 12.5 23.8
Ry 4.1 21.7 25.8
_at_
18
Alternate Method for Adding Non-perpendicular
Vectors

• Consider a vector triangle with angles A, B, and
  C with opposites sides labeled a, b, and c

A
c
b
C
B
a
19
Alternate Method for Adding Non-perpendicular
Vectors

• Cosine law can be used to find magnitude of
  vector c if magnitudes and directions of a and b
  are known
• Angle between a and b can be found using simple
  geometry
• Sine law can be used to find direction of vector c

20
Cosine Law

• Useful if two sides (a and b) and the angle
  between them (C) are known
• Similar to Pythagorean Theorem with a correction
  factor for lack of right angle

21
Sine Law

• Useful when one angle and its opposite side are
  known along with one other side or angle