VECTORS
Do you remember the difference
between a scalar and a vector?
Scalars are quantities which are fully
described by a
magnitude alone.
Vectors are quantities which are fully
described by both a
magnitude and
a direction.
1. The length of the line represents the
magnitude and the arrow indicates the
direction.
2. The magnitude and direction of the vector is
clearly labeled.
Scaling!!!
The magnitude of a vector in a scaled vector
diagram is depicted by the length of the arrow.
The arrow is drawn precisely to length in
accordance with a chosen scale.
Compass Coordinate System
Direction!!!
Sometimes vectors will be directed
due East or due North. However we
will encounter vectors in all sorts of
directions and be forced to find the
angle!
Compass Coordinate System
Use the same scale for all vector magnitudes
•
•
•
•
Δx = 30 m @ 20º E of N
V = 20 m/s @ 30º W of N
a = 10 m/s2@ 40º W of S
F = 50 N @ 10º S of E W
N
E
To Draw direction:
Ex. 20º E of N: Start w/ North and go 20° East
Navigational System?
S
All these planes have the same reading on
their speedometer. (plane speed not speed
with respect to the ground (actual speed)
What
factor is
affecting
their
velocity?
A
B
C
Easy Adding…
The resultant is the
vector sum of two or
more vectors.
1. Select an appropriate scale (e.g., 1 cm = 5 km)
2. Draw and label 1st vector to scale.
*The tail of each consecutive vector begins at the head of
the most recent vector*
3. Draw and label 2nd vector to scale starting at the head of
the 1st vector.
4. Draw the resultant vector (the summative result of the
addition of the given vectors) by connecting the tail of the
1st vector to the head of the 2nd vector. (initial to final pt.)
5. Determine the magnitude and direction of the resultant
vector by using a protractor, ruler, and the indicated
scale; then label the resultant vector.
120 km/h
20 km/h
80 km/h
100 km/h
=
100 km/h
=
20km/h
A. Tailwind
(with the wind)
B. Headwind
(against the wind)
C. 90º crosswind
Resultant
80 km/h
100 km/h
60 km/h
Using a ruler and your
scale, you can determine
the magnitude of the
resultant vector. Or you
could use the
Pythagorean Theorem.
Then using a protractor,
you can measure the
direction of the resultant
vector. Or you could use
trigonometry to solve for
the angle.
1. Find the resultant force vector of the two forces
below.
25 N due East, 45 N due South
25 N, East
Decide on
a scale!!! 31º51E Nof S
51 N
59º S of E
45 N, South
An airplane is flying 200mph at 50o N of E. Wind velocity
is 50 mph due S. What is theN velocity of the plane?
180
o
0
270
Scale: 50 mph = 1 inch
o
o
An airplane is flying 200mph at 50o N of E. Wind velocity
is 50 mph due S. What is theNvelocity of the plane?
W
E
S
Scale: 50 mph = 1 inch
An airplane is flying 200mph at 50o N of E. Wind velocity
N
is 50 mph due S. What is the velocity of the plane?
50 mph
200 mph
W
E
VR = 165 mph
@ 40° N of E
S
Scale: 50 mph = 1 inch
2. Find the resultant velocity vector of the two
velocity vectors below.
700 m/s @35 degrees E of N; 1000 m/s @ 30 degrees
N of W
V2
Vr
V1
Intro to Vectors Warm-up
A bear walks one mile
south, then one mile
west, and finally walks
one mile north. After
his brisk walk, the bear
ends back where he
started.
What color is the
bear???
In what direction is the leash pulling
on the dog?
What would happen to the upward
and rightward Forces if the Force on
the chain were smaller?
1) Find the resultant
of the two vectors
24.99 N
Magnitude:__________
34.90º N of W
Direction:___________
Vector #1 = 20.5 N West
Vector #2 = 14.3 N North
Vector Diagram
V2
VR=?
V1
2) Find the component of the
resultant = 255m 27º South of East
115.8 m Direction__________
South (-)
Vector # 1 _______
East (+)
227.2 m Direction__________
Vector # 2________
Vector Diagram:
Conventions:
V2
Vr
+
V1
-
+
-
Skip Practice: Find FR = Fnet =?
200 N due South,
100 N at 40º N of W
1. Draw vector diagram. (Draw axis)
2. Resolve vectors into components using trig:
Vadj = V cos θ
Vopp = V sin θ
3. Sum x and y components:
ΣVxi
ΣVyi
4. Redraw!! Determine resultant vector using
Pythagorean’s Theorem and trig:
Magnitude= √(Σ Vxi)² + (Σ Vyi)²
Direction Action: θ = tan-1(opp/adj)
Answer: Fnet = N
@ ˚ W of S
An airplane flies at an engine speed of 100 m/s
at 50º W of S into a wind of 30 m/s at 200 E of
N. What is the airplane’s resultant velocity?
Solve using the components method!!
How far has the plane traveled after 1 hr?
a) Km
b) Mph
Answer: 75.52 m/s @ 28.54˚ S of W
271 km or 168.89 miles per 1 hour
You Try!!!
A motor boat traveling 4.0 m/s, East
encounters a current traveling 3.0 m/s,
North.
a. What is the resultant velocity of the motor boat?
b. If the width of the river is 80 meters wide, then how
much time does it take the boat to travel shore to
shore?
c. What distance downstream does the boat reach the
opposite shore?
Another look, from a
different perspective
Non-Collinear Vectors
When 2 vectors are perpendicular, you must use the
Pythagorean theorem.
A man walks 95 km, East then 55
km, north. Calculate his
The hypotenuse in Physics is
called the RESULTANT
Finish
.
RESULTANTDISPLACEMENT
.
c2  a2  b2  c  a2  b2
55 km, N
Horizontal Component
Vertical
Component
c  Resultant  952  552
c  12050  109.8 km
95 km,E
Start
The LEGS of the triangle are called the COMPONENTS
BUT……what about the
direction?
In the previous example, DISPLACEMENT was asked for and
since it is a VECTOR we should include a DIRECTION on
our final answer.
N
W of N
E of N
N of E
N of W
N of E
W
NOTE: When drawing a right triangle that
conveys some type of motion, you MUST draw
your components HEAD TO TOE.
S of W
W of S
S of E
E of S
S
E
BUT…..what about the VALUE of the
angle???
Just putting North of East on the answer is NOT specific enough for the
direction. We MUST find the VALUE of the angle.
109.8 km
 N of E
95 km,E
55 km, N
To find the value of the
angle we use a Trig
function called
TANGENT.
opposite side 55
Tan 

 0.5789
adjacent side 95
  Tan 1 (0.5789)  30
So the COMPLETE final answer is : 109.8 km, 30 degrees North of East
What if you are missing a component?
Suppose a person walked 65 m, 25 degrees East of North. What were
his horizontal and vertical components?
H.C. = ?
The goal: ALWAYS MAKE A RIGHT
TRIANGLE!
V.C = ?
25
65 m
To solve for components, we often use
the trig functions since and cosine.
adjacent side
hypotenuse
adj  hyp cos 
cosine  
opposite side
hypotenuse
opp  hyp sin 
sine  
adj  V .C.  65 cos 25  58.91m, N
opp  H .C.  65 sin 25  27.47m, E
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.
- =
12 m, W
-=
6 m, S
14 m, N
20 m, N
35 m, E
R

23 m, E
R  14 2  232  26.93m
14 m, N
14
 .6087
23
  Tan 1 (0.6087)  31.3
Tan 
23 m, E
The Final Answer: 26.93 m, 31.3 degrees NORTH or EAST
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.
Rv  82  152  17 m / s
8.0 m/s, W
15 m/s, N
Rv

8
Tan 
 0.5333
15
  Tan 1 (0.5333)  28.1
The Final Answer : 17 m/s, @ 28.1 degrees West of North
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.
adjacent side
cosine  
hypotenuse
adj  hyp cos 
H.C. =?
32
opposite side
sine  
hypotenuse
opp  hyp sin 
V.C. = ?
63.5 m/s
adj  H .C.  63.5 cos 32  53.85 m / s, E
opp  V .C.  63.5 sin 32  33.64 m / s, S
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.
1500 km
adjacent side
hypotenuse
V.C.
adj  hyp cos 
cosine  
opposite side
hypotenuse
opp  hyp sin 
sine  
40
5000 km, E
H.C.
adj  H .C.  1500 cos 40  1149.1 km, E
opp  V .C.  1500 sin 40  964.2 km, N
5000 km + 1149.1 km = 6149.1 km
R  6149.12  964.2 2  6224.14 km
964.2
 0.157
6149.1
  Tan 1 (0.364)  8.91
Tan 
R
964.2 km

6149.1 km
The Final Answer: 6224.14 km @
8.91 degrees, North of East
Add vectors that are NOT
perpendicular
• If the original displacement vectors do not
form a right triangle
– 1. Resolve each vector into its x- and ycomponents
– 2. Find the sum of the x- and y-components
– 3. Use the Pythagorean Theorem to find the
magnitude of the resultant
– 4. Use the tangent function to find the direction
of the resultant
Adding non-perpendicular vectors
Adding non-perpendicular vectors
Adding non-perpendicular vectors
Practice #1
• A hiker walks 27.0 km from her base camp
at 35 south of east. The next day, she
walks 41.0 km in a direction 65 north of
east and discovers a forest ranger’s tower.
Find the magnitude and direction of her
resultant displacement between the base
camp and the tower.
Answer ~ 45 km at 29 degrees N of E