The document describes scalar and vector quantities in physics. Scalars have magnitude only, while vectors have both magnitude and direction. The document discusses how to add and subtract vectors by combining their components and provides examples. It also covers using trigonometry to calculate resultant vectors when vectors are perpendicular, including finding the direction angle.

1. VECTORS
Honors Physics

2. SCALAR
A SCALAR quantity
is any quantity in
physics that has
MAGNITUDE ONLY
Number value
with units
Scalar
Example Magnitude
Speed 35 m/s
Distance 25 meters
Age 16 years

3. VECTOR
A VECTOR quantity
is any quantity in
physics that has
BOTH MAGNITUDE
and DIRECTION
Vector
Example
Magnitude and
Direction
Velocity 35 m/s, North
Acceleration 10 m/s2, South
Force 20 N, East
An arrow above the symbol
illustrates a vector quantity.
It indicates MAGNITUDE and
DIRECTION

4. VECTOR APPLICATION
ADDITION: When two (2) vectors point in the SAME direction, simply
add them together.
EXAMPLE: A man walks 46.5 m east, then another 20 m east.
Calculate his displacement relative to where he started.
66.5 m, E
MAGNITUDE relates to the
size of the arrow and
DIRECTION relates to the
way the arrow is drawn
46.5 m, E + 20 m, E

5. VECTOR APPLICATION
SUBTRACTION: When two (2) vectors point in the OPPOSITE direction,
simply subtract them.
EXAMPLE: A man walks 46.5 m east, then another 20 m west.
Calculate his displacement relative to where he started.
26.5 m, E
46.5 m, E
-
20 m, W

6. Vectors are always added head to tail
For example draw what you think it would look like if I
added these two vectors together: A+B. Then draw what
you think A-B would look like.
A B
A+B
A-B

7. You try on the whiteboard
A B
C
a.A+C
b.B+C
c.A-B
d.C-B
e.A+B

8. NON-COLLINEAR VECTORS
When two (2) vectors are PERPENDICULAR to each other, you must
use the PYTHAGOREAN THEOREM
Example: A man travels 120 km east
then 160 km north. Calculate his
resultant displacement.
120 km, E
160 km, N
the hypotenuse is
called the RESULTANT
HORIZONTAL COMPONENT
VERTICAL
COMPONENT
S
T
A
R
T
FINISH
c2
 a2
 b2
 c  a2
 b2
c  resultant  120
 
2
 160
 
2
 
c  200km

9. WHAT ABOUT DIRECTION?
In the example, DISPLACEMENT asked for and since it is a VECTOR quantity,
we need to report its direction.
N
S
E
W
N of E
E of N
S of W
W of S
N of W
W of N
S of E
E of S
NOTE: When drawing a right triangle that
conveys some type of motion, you MUST
draw your components HEAD TO TOE.
N of E

11. NEED A VALUE – ANGLE!
Just putting N of E is not good enough (how far north of east ?).
We need to find a numeric value for the direction.
N of E
160 km, N
120 km, E
To find the value of the
angle we use a Trig
function called TANGENT.

Tan 
opposite side
adjacent side

160
120
1.333
  Tan1
(1.333)  53.1o

200 km
So the COMPLETE final answer is : 200 km, 53.1 degrees North of East

12. What are your missing
components?
Suppose a person walked 65 m, 25 degrees East of North. What
were his horizontal and vertical components?
65 m
25
H.C. = ?
V.C = ?
The goal: ALWAYS MAKE A RIGHT
TRIANGLE!
To solve for components, we often use
the trig functions since and cosine.
E
m
C
H
opp
N
m
C
V
adj
hyp
opp
hyp
adj
hypotenuse
side
opposite
hypotenuse
side
adjacent
,
47
.
27
25
sin
65
.
.
,
91
.
58
25
cos
65
.
.
sin
cos
sine
cosine















13.  Draw the vector diagram
 Combine ‘like’ vectors
 Determine the resultant AND the angle of direction
A bear, searching for food wanders 35 meters east
then 20 meters north. Frustrated, he wanders another
12 meters west then 6 meters south. Calculate the
bear's displacement.

14. Example
A bear, searching for food wanders 35 meters east then 20 meters north.
Frustrated, he wanders another 12 meters west then 6 meters south. Calculate
the bear's displacement.
35 m, E
20 m, N
12 m, W
6 m, S
- =
23 m, E
- =
14 m, N
23 m, E
14 m, N

3
.
31
)
6087
.
0
(
6087
.
23
14
93
.
26
23
14
1
2
2








Tan
Tan
m
R


The Final Answer: 26.93 m, 31.3 degrees NORTH or EAST
R


15.  Draw the vector diagram
 Combine ‘like’ vectors (if necessary)
 Determine the resultant AND the angle of direction
A boat moves with a velocity of 15 m/s, N in a river which
flows with a velocity of 8.0 m/s, west. Calculate the boat's
resultant velocity with respect to due north.

16. Example
A boat moves with a velocity of 15 m/s, N in a river which
flows with a velocity of 8.0 m/s, west. Calculate the
boat's resultant velocity with respect to due north.
15 m/s, N
8.0 m/s, W
Rv 

1
.
28
)
5333
.
0
(
5333
.
0
15
8
/
17
15
8
1
2
2








Tan
Tan
s
m
Rv


The Final Answer : 17 m/s, @ 28.1 degrees West of North

17. A plane moves with a velocity of 63.5 m/s at 32
degrees South of East. Calculate the plane's
horizontal and vertical velocity components.
 Draw the vector diagram
 Combine ‘like’ vectors (if necessary)
 Determine the resultant AND the angle of direction

18. Example
A plane moves with a velocity of 63.5 m/s at 32 degrees South of East. Calculate
the plane's horizontal and vertical velocity components.
63.5 m/s
32
H.C. =?
V.C. = ?
S
s
m
C
V
opp
E
s
m
C
H
adj
hyp
opp
hyp
adj
hypotenuse
side
opposite
hypotenuse
side
adjacent
,
/
64
.
33
32
sin
5
.
63
.
.
,
/
85
.
53
32
cos
5
.
63
.
.
sin
cos
sine
cosine















19. A storm system moves 5000 km due east, then shifts course at
40 degrees North of East for 1500 km. Calculate the storm's
resultant displacement.
 Draw the vector diagram
 Combine ‘like’ vectors (if necessary)
 Determine the resultant AND the angle of direction

20. Example
A storm system moves 5000 km due east, then shifts course at 40
degrees North of East for 1500 km. Calculate the storm's
resultant displacement.
N
km
C
V
opp
E
km
C
H
adj
hyp
opp
hyp
adj
hypotenuse
side
opposite
hypotenuse
side
adjacent
,
2
.
964
40
sin
1500
.
.
,
1
.
1149
40
cos
1500
.
.
sin
cos
sine
cosine














5000 km, E
40
1500 km
H.C.
V.C.
5000 km + 1149.1 km = 6149.1 km
6149.1 km
964.2 km
R


R  6149.12
 964.22
 6224.2km
Tan 
964.2
6149.1
 0.157
  Tan1
(0.157)  8.92o
The Final Answer: 6224.2 km @ 8.92
degrees, North of East