Physics B
Classwork: Vectors
Name:________________
Period:________
Magnitude of Vectors
Vectors
Definition:
The best way to determine the magnitude of a vector is
to measure its length.
The length of the vector is proportional to the
magnitude (or size) of the quantity it represents.
Examples:
Sample Problem
Scalars
Definition:

If vector A represents a displacement of three
miles to the north, then what does vector B
represent? Vector C?
Examples:
B
A
This is how you draw a vector…
C
Direction of Vectors
Vector direction is the direction of the arrow, given
by an angle.
Equal Vectors
Equal vectors have the same length and direction, and
represent the same quantity (such as force or velocity).
Draw several equal vectors.
Vector angle ranges
Inverse Vectors
Inverse vectors have the same length, but opposite
direction.
Draw a set of inverse vectors.
Right Triangles
What quadrant is the angle of Vector B in?
Draw a right triangle, designating the opposite,
adjacent, and hypotenuse.
B
State the Pythagorean Theorem.
What would be the exact angle, and how would you
determine it?
2/17/2016
1
Bertrand
List the trig functions you need for Physics B.
Vectors: y-component
How do you determine the y-component of a vector?
Vectors: angle
List the inverse trig functions you need.
How do you determine the angle a vector makes with
the x-axis?
Sample problem
A surveyor stands on a riverbank directly across the
river from a tree on the opposite bank. She then
walks 100 m downstream, and determines that the
angle from her new position to the tree on the
opposite bank is 50o. How wide is the river, and how
far is she from the tree in her new location?
Vectors: magnitude
How do you determine the magnitude of a vector?
Sample Problem
You are driving up a long inclined road. After 1.5 miles
you notice that signs along the roadside indicate that
your elevation has increased by 520 feet.
a) What is the angle of the road above the horizontal?
Sample problem
You are standing erect at the very top of a tower
and notice that in order to see a manhole cover on
the ground 50 meters from the base of the tower,
you must look down at an angle 75o below the
horizontal. If you are 1.80 m tall, how high is the
tower?
b) How far do you have to drive to gain an
additional 150 feet of elevation?
Sample Problem
Find the x- and y-components of the following
vectors
R = 175 meters @ 95o
Vectors: x-component
How do you determine the x-component of a vector?
v = 25 m/s @ -78o
a = 2.23 m/s2 @ 150o
2/17/2016
2
Bertrand
Graphical Addition of Vectors
1)
2)
3)
4)
Add vectors A and B graphically by drawing
them together in a head to tail arrangement.
Draw vector A first, and then draw vector B
such that its tail is on the head of vector A.
Then draw the sum, or resultant vector, by
drawing a vector from the tail of A to the head
of B.
Measure the magnitude and direction of the
resultant vector.
b)
Find A + B + C graphically.
Practice Graphical Addition
The Resultant and the Equilibrant
c) Fnd A – B graphically.
Definition of Resultant:
Definition of Equilibrant:
Graphical Subtraction of Vectors
1)
2)
3)
Subtract vectors A and B graphically by
adding vector A with the inverse of vector B
(-B).
First draw vector A, then draw -B such that
its tail is on the head of vector A.
The difference is the vector drawn from the
tail of vector A to the head of -B.
Practice Graphical Subtraction
Vector Addition Lab
1.
2.
3.
4.
5.
Sample Problem
Vector A points in the +x direction and has a magnitude
of 75 m. Vector B has a magnitude of 30 m and has a
direction of 30o relative to the x axis. Vector C has a
magnitude of 50 m and points in a direction of -60o
relative to the x axis.
a) Find A + B graphically.
6.
7.
8.
Attach spring scales to force board such that they all
have different readings.
Slip graph paper between scales and board and
carefully trace your set up.
Record readings of all three spring scales.
Detach scales from board and remove graph paper.
On top of your tracing, draw a force diagram by
constructing vectors proportional in length to the scale
readings. Point the vectors in the direction of the
forces they represent. Connect the tails of the vectors
to each other in the center of the drawing.
On a separate sheet of graph paper, add the three
vectors together graphically. Identify your resultant, if
any.
Did you get a resultant? Did you expect one?
You must have a separate set of drawings for each
member of your lab group, so work efficiently
Component Addition of Vectors
1. Resolve each vector into its x- and y-components.
Ax = Acos
Ay = Asin
Bx = Bcos
By = Bsin
Cx = Ccos
Cy = Csin
2. Add the x-components (Ax, Bx, etc.) together to get Rx
and the y-components (Ay, By, etc.) to get Ry.
3. Calculate the magnitude of the resultant with the
Pythagorean Theorem (R = Rx2 + Ry2).
4. Determine the angle with the equation  = tan-1 Ry/Rx.
2/17/2016
3
Bertrand
Sample Problem
In a daily prowl through the neighborhood, a cat makes
a displacement of 120 m due north, followed by a
displacement of 72 m due west. Find the magnitude
and displacement required if the cat is to return home.
Do this by the component method.
c)
You want to paddle straight across the river,
from the south to the north. At what angle to
you aim your boat relative to the shore?
Assume east is 0o.
Sample Problem
You are flying a plane with an airspeed of 400 mph. If
you are flying in a region with a 80 mph west wind, what
must your heading be to fly due north?
Sample Problem
If the cat in the previous problem takes 45 minutes to
complete the first displacement and 17 minutes to
complete the second displacement, what is the
magnitude and direction of its average velocity during
this 62-minute period of time? Do this by the
component method.
Relative Motion
Relative motion problems are difficult to do unless one
applies vector addition concepts.
Define a vector for a swimmer’s velocity relative to the
water, and another vector for the velocity of the water
relative to the ground. Adding those two vectors will
give you the velocity of the swimmer relative to the
ground.
Free Response Practice Problem
a)
Practice Problems
You are paddling a canoe in a river that is flowing at 4.0
mph east. You are capable of paddling at 5.0 mph.
a) If you paddle east, what is your velocity
relative to the shore?
b)
b) On the reverse side of your graph paper,
check your graphical addition by doing
a component addition. Again, clearly
indicate the magnitude and direction of
the resultant.
If you paddle west, what is your velocity
relative to the shore?
2/17/2016
Graphically determine the resultant of
the following vectors. Use your own
graph paper. Clearly indicate the
magnitude and direction of the resultant
on your drawing.
A: 60 m at 40o
B: 90 m at 175o
o
C: 50 m at -45
4
Bertrand