• 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
Let’s say you
walk 10 m
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
• 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
use LAW OF SINES or LAW OF COSINES
• 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.
• 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
they’re not drawn
in a tip-to-tail
Let’s say you want
to measure the
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.
rest in the
(the hole in
and one of
Let’s move the with the
arrows. How would marked
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
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,
m/s2 or for
force in N but
• Again, since it’s a right triangle, we can
• 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
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
c2 = a2 + b2 – 2ab cos C c
Use this website to do a quick calculation: 2.3 m
Use the rest of the class time to
work in small, quiet groups to draw
and calculate the resultants.