Introduction to Vectors - Download as PowerPoint by hcj


									• Scalar – a quantity with magnitude only
  Speed: “55 miles per hour”
• Temperature: “22 degrees Celsius”

• Vector – a quantity with magnitude and
• Velocity: “25 m/s Northwest”
• Acceleration: “4.8 m/s2 at 90 degrees”
• Component – one of the vectors given
  in the problem
• Resultant – the “net” vector
• Concurrent Vectors – vectors acting on
  the same point at the same time
• Equilibrant – vectors that produce
  equilibrium; it is equal in magnitude
  and opposite in direction to the
• Examples:
  Let’s say you
  walk 10 m
  northeast and
  turn and walk
• Problem: Hurricane Frances is traveling
  at 8 mph West. A weather front
  approaches Frances at 20 mph
  Northeast. Find the resultant direction
  of Hurricane Frances. (“net”)
• (OK – so it’s more complicated than
• What to do?
• Go to:
• (Make sure the volume is turned UP!)

• Watch the video. What does this video
  show? (Watch it more than once if
• That the “river” was the paper being
  pulled at a constant rate?
• That if the “boat” left perpendicularly to
  the “shore” it would end up
• That the instructor must have
  calculated the proper angle needed
  based on the rate of “flow of the river”
  before he released the second “boat”?
• This is vital information for pilots, ship
  navigators, athletes
• A quarterback uses vectors to throw a
  ball to a receiver that is running to
  make a touchdown (but he probably
  doesn’t think of it as a vector)
• Name some other examples in sports
  or other aspects of your life
• Graphically – vectors are drawn using a ruler
  (measurements are done to scale) and a
  protractor (direction is noted as, for example,
  20 degrees north of east)

• Analytically – trig is used; right triangles use
  the Pythagorean Theorem (a2 + b2 = c2) and
  SOH CAH TOA. Triangles that are not right
• In fact, you will be taking one quiz in
  which you will have only a protractor
  and one quiz in which you will have
  only a calculator.
• Resultant – is the “net” direction, or
  force, or acceleration, etc.
• Use a protractor to draw a line going in
  the direction stated. The LENGTH of
  the line indicates the MAGNITUDE of
  the direction, force, etc.
• The DIRECTION of the vector is in the
  stated direction and is carefully
  measured using the protractor
• A plane’s engine pulls the plane 700m
  to the north. There is a strong wind
  pushing the plane 200 m to the west.

• Let’s solve this problem graphically…
• First, carefully draw an arrow pointing
  north that has a magnitude of 700 units.

                           Then, draw an arrow
                           pointing west that has a
                           magnitude of 200 units.
                         Notice that we have used a “tip-to-
   Now you have to
                         tail method” to draw vectors. Two
   draw the Resultant,
                         tails exist only where the resultant
   which shows the
                         touches the “first” vector.
   “net” magnitude.
• It is necessary to draw vectors using
  the Tip-To-Tail Method. That means
  that the tip of one vector can only
  touch the tail of a second vector in the
  final vector diagram.
• See how this is done at this site, where
  the resultant is shown in RED:
Now that you know the magnitude of the
  resultant, you must report the direction
  of the resultant. (Remember – that’s
  what makes it a vector!)
Put your protractor’s “origin” at the
  intersection between the original vector
  and the resultant.
Find the angle and state the angle as, ie,
  “22 degrees N of E”
                                        You will probably
                                        note that these
                                        can’t officially be
                                        vectors since
                                        they’re not drawn
                                        in a tip-to-tail

Let’s say you want
to measure the
angle between
these two arrows…
 You could subtract 90.0 from 127.5 to get 37.5
 degrees. You would record this as 37.5 degrees
 East of North.
                                                           The ends
                                                           of each
                                                           arrow must
                                                           rest in the
                                                           “origin” of
                                                           (the hole in
                                                           the plastic)
                                                           and one of
                                                           the vectors
                                                           must align
          Let’s move the                                   with the
          arrows. How would                                marked
                                                           black line.
          you measure this

You could subtract 0.0 from 78.5 to get 78.5 degrees.
You could state that this is 78.5 degrees West of South.
• Let’s take that first example of the
  700 m displacement N and the 200 m
  displacement W.
 Since this makes a Right Triangle, we
 can use the Pythagorean Theorem to
 solve for the resultant.
           a2 + b2 = c2

   Therefore,        7002 + 2002 = c2

                     And    c = 728 m
A vector diagram
  can be drawn
  in m, for
  velocity in m/s,
  acceleration in
  m/s2 or for
  force in N but
  each vector
  sides have
• Again, since it’s a right triangle, we can
  use Trig.
• Since we know the opp and adj, and we
  know that     tan q = opp / adj,
  calculate this, too.
• For questions 5-8 on the “Vectors I”
  handout, 2 vectors act on a single
  point, like this…
• …you must remember the tip-to-tail
  method so you will have to slide one
  vector to the end of another as shown
  on the site you saw earlier.
  Remember? Return to that site if
  needed. Here’s the link again.
 You Must Pick One Vector to
Slide Onto the End of the Other
• Let’s say you
  have one
  vector that is
  7.8 m , one
  that is 2.3 m
  and you have
  to find the
• If you know the
  sides (7.8 and                              Here are the sides (lower case)
  2.3) and Angle C
  (110o) then you
  must use the                                          7.8 m

                                                         a          Here are the
                                                                    (upper case)
c2 = a2 + b2 – 2ab cos C                           c
                                                                 C 110
Use this website to do a quick calculation:                         2.3 m
                                                             A                b
 Use the rest of the class time to
work in small, quiet groups to draw
   and calculate the resultants.

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