Vectors represent magnitude and direction. Vectors can be named like a ray,
or in
bold with one letter in bold, u (or in handwritten text).
The magnitude of vector is the size of a vector often representing force or velocity.
The direction of a vector is an angle measurement where 0° is to the right on the
horizontal.
Directio
n
90
135°
°
45
°
0°
180°
225
°
315°
270°
I. Model Problems
In the following problem you will learn to show vector addition using the tail-to-tip
method.
Find
.
v
75°
u
Translate v. Slide v
along u so that the tail
of v is at the tip of u .
v
75°
v
75°
u
Draw the resultant
vector,
, which
starts at the tail of u
and ends at the tip of v.
v
75°
u
!"#$%&'()*#+"$(
Name: ___________________________________
Vectors are simply anything with a magnitude (size) and direction. They are
represented as arrows.
In the world of physics, the reason we deal with things in vectors is because it makes
it easy to keep track of multiple things at once.
Imagine a bunch of teenage girls on the volley ball team fighting over the ball. If
there are twelve girls pulling on the volley ball from different directions with
different strengths all at the same time, how could you find out the overall movement
of the ball? It would be nearly impossible without vectors.
See, vectors have a bit of a super power. They can be broken down into components.
Take a look at the vector below.
,-#%./%0"0$(
1-#%./%0"0$(
It has an “x” component and a “y” component. If you imagine each vector like a
triangle, the x-component is the side to side leg, and the y-component is the up and
down leg of the triangle. In order to add multiple vectors together, all you have to do
is break each vector apart into its components, then add all the components together.
"(
234(
56"(7"#$%&(*8%7"9(7"#$%&(:!;(6*'(*(.*<0=$>?"(%@(ABC(56"(1-#%./%0"0$(:!1:(='(
"D>*E($%($6"(#%'=0"(%@($6"(*0<E"9($=."'($6"(.*<0=$>?"(%@($6"(7"#$%&C(F=.=E*&E,9(
$6"(,-#%./%0"0$(:!,;(='("D>*E($%($6"('=0"(%@($6"(*0<E"($=."'($6"(7"#$%&C(
!1(G(AB#%'H234I(
!,(G(AB'=0H234I(
I I. V ector Basics
1. What is the magnitude and direction of
2. What is the magnitude and direction of
B
R
4.5 lb
8.5
28°
A
110°
T
3. What is the magnitude and direction of
.
4. Sketch the resultant vector
J
70°
v
12 lb
20°
K
u
.
5. Sketch the resultant vector
u
115°
7. What is the magnitude and direction of
the resultant in the sketch below
13
8. What is the magnitude and direction of
the resultant in the sketch below
12
53°
6
110°
27°
17°
12
10
11
9. What is the magnitude and direction of
the resultant in the sketch below
9
53°
15
90°
12
37°
v
85°
u
v
.
6. Sketch the resultant vector
60°
103°
4
I I I. A ddition of V ectors
10. Vector u has a magnitude of 20 and a
direction of 0°. Vector v has a
magnitude of 40 and a direction of 60°.
Find the magnitude and direction of the
resultant to the nearest whole number.
11. Vector u has a magnitude of 15 and a
direction of 0°. Vector v has a
magnitude of 18 and a direction of 70°.
Find the magnitude and direction of the
resultant to the nearest whole number.
18
40
70°
60°
15
20
12. Vector u has a magnitude of 24 and a
direction of 0°. Vector v has a
magnitude of 40 and a direction of
115°. Find the magnitude and direction
of the resultant to the nearest whole
number.
(continued on next page)
I V . Find the magnitude of the resultant vector when two forces are applied to an
object.
13. Two forces with magnitudes of 20
pounds and 14 pounds and an angle of
55° between them are applied to an
object. Find the magnitude of the
resultant vector to the nearest whole
number.
20 lb
55° 14 lb
15. Two forces with magnitudes of 70
pounds and 40 pounds and an angle of
130° between them are applied to an
object. Find the magnitude of the
resultant vector to the nearest whole
number.
17. Two forces with magnitudes of 62
pounds and 62 pounds and an angle of
145° between them are applied to an
object. Find the magnitude of the
resultant vector to the nearest whole
number.
14. Two forces with magnitudes of 48
pounds and 65 pounds and an angle of
80° between them are applied to an
object. Find the magnitude of the
resultant vector to the nearest whole
number.
65 lb
80°
48 lb
16. Two forces with magnitudes of 77
pounds and 45 pounds and an angle of
43° between them are applied to an
object. Find the magnitude of the
resultant vector to the nearest whole
number.
!"#$%&'()*#+"$(
18:
1: In the fictitious city of Metropolis, streets run east-west while avenues run northsouth. The separation between centers for both the streets and the avenues is 100 m,
resulting in square city blocks. Superman is located at 33rd Street and 3rd Avenue
while Lois Lane is located north and east of Superman at 45th Street and
12th Avenue.
19:
1: How far would Lois Lane have to walk to reach Superman?
2: How far would Superman have to fly to reach Lois Lane?
20:
21:
3: In what direction should Superman fly to reach Lois Lane as quickly as possible?
22:
4: Helen is parasailing. She sits in a seat harness which is attached by a tow rope to a
speedboat. The rope makes an angle of 51° with the horizontal and has a tension of
350 N. Determine the horizontal and vertical components of the tension force.
23-25:
5: For each collection of listed forces, determine the vector sum or the net force.
Set A
58 N, right
42 N, left
98 N, up
98 N, down
Set B
14 N, left
16 N, up
16 N, down
Set C
12 N, up
8 N, down
26:
6: A pack of three Artic wolves are fighting over the carcass of a dead polar bear. A top
view of the magnitude and direction of the three forces is shown in the diagram to the
right. Determine the resultant or net force acting upon the carcass.
27: A force of 120 N is found to be at a 45˚ angle with another force of 55 N. What
force should be added to make the resulting net force 0 N?
28: What does the “Magnitude” of a vector mean?
29: What is a “resultant” vector?
30: A student adds 10 vectors together and gets a resultant vector of 20 N at an
angle of 60˚. The student then adds the 10 vectors together in reverse order, starting
with the 10th and ending with the 1st. What will the resultant vector be in this case?