Vectors and Scalars
A SCALAR is ANY quantity in physics
that has MAGNITUDE, but NOT a
direction associated with it.
Magnitude – A numerical value with
units.
Scalar
Example
Magnitude
Speed
20 m/s
Distance
10 m
Age
15 years
Heat
1000
calories
Number of
horses
behind the
school
I guess: 12
A VECTOR is ANY quantity in
physics that has BOTH
MAGNITUDE and DIRECTION.
Vector
Velocity
Magnitude &
Direction
20 m/s, N
Acceleration 10 m/s/s, E
Force
5 N, West
A picture is worth a thousand word, at least they say so.
Vectors are typically illustrated by drawing an ARROW above the symbol.
The arrow is used to convey direction and magnitude.
250
Tail
H
L Head
length = magnitude
6 cm
250 above x-axis = direction
displacement x = 6 cm, 250
v
The length of the vector, drawn
to scale, indicates the
magnitude of the vector quantity.
the direction of a vector is the
counterclockwise angle of rotation which
that vector makes with due East or x-axis.
A resultant (the real one) velocity is sometimes
the result of combining two or more velocities.
A small plane is heading south at speed of 200 km/h
(If there was no wind plane’s velocity would be 200 km/h south)
1. The plane encounters a
tailwind of 80 km/h.
2. It’s Texas: the wind changes
direction suddenly 1800.
Velocity vectors are now in
opposite direction.
80 km
200 km h 80 km h
e
200 km h
280 km
h
resulting velocity relative
to the ground is 280 km/h
200 km h
h
e
120 km h
Flying against a 80 km/h wind, the
plane travels only 120 km in one
hour relative to the ground.
You can use common sense to find resulting velocity of
the plane in the case of tailwind and headwind, but if the
wind changes direction once more and wind velocity is
now at different angle, combining velocities is not any
more trivial. Then, it’s just right time to use vector algebra.
3. The plane encounters a crosswind of 80 km/h.
Will the crosswind speed up the plane, slow it down, or have no effect?
80 km
h
200 km h
HELP: In one hour plane will move 80 km east and 200 km south, So it will cover
more distance in one hour then if it was moving south only at 200 km/h.
To find that out we have to add these two vectors.
The sum of these two vectors is called RESULTANT.
The magnitude of resultant velocity (speed v)
can be found using Pythagorean theorem
80 km h
200 km h
v= v12 +v 22 = (200km/h)2 + (80km/h)2 = 46400km2 /h2
RESULTANT
RESULTANT VECTOR
(RESULTANT VELOCITY)
v = 215 km/h
So relative to the ground, the plane
moves 215 km/h southeasterly.
Very unusual math, isn’t it? You added 200 km/h and 80 km/h and
you get 215 km/h. 1 + 1 is not necessarily 2 in vector algebra.
200 km h
280 km h
120 km h
215 km h
180 km h
If the air velocity was not at the right angle to the plane velocity, you intuitively
know that the speed of the plane would be different. So we are coming to the
surprising result. 200 + 80 can be almost anything if 200 and 80 have direction.
Vector Addition: 6 + 5 = ?
Till now you naively thought that 6 + 5 = 11.
In vector algebra
Not so fast
6 + 5 can be 10 and 2, and 8, and…
When two forces are acting on you, for example 5N and 6N, the resultant
force, the one that can replace these two having the same effect, will
depend on directions of 5N force and 6N force. Adding these two vectors
will not necessarily result in a force of 11N.
The rules for adding vectors are different than the rules for adding two
scalars, for example 2kg potato + 2kg potatos = 4 kg potatoes. Mass
doesn’t have direction.
Vectors are quantities which include direction. As such, the addition of two
or more vectors must take into account their directions.
There are a number of methods for carrying out
the addition of two (or more) vectors.
The most common methods are: "head-to-tail"
and “parallelogram” method of vector addition.
We’ll first do head-to-tail method, but before that, we
have to introduce multiplication of vector by scalar.
Two vectors are equal if they have the same magnitude
and the same direction.
This is the same vector. It doesn’t matter where it is. You
can move it around. It is determined ONLY by magnitude
and direction, NOT by starting point.
Multiplying vector by a scalar
Multiplying a vector by a
scalar will ONLY CHANGE
its magnitude.
A
2A
3A
½A
Multiplying vector by 2 increases its magnitude
by a factor 2, but does not change its direction.
One exception:
Multiplying a vector by “-1” does not
change the magnitude, but it does
reverse it's direction
Opposite vectors
–A
A
-A
– 3A
Vector addition - head-to-tail method
vectors: 6 units,E + 5 units,300
examples:
v – velocity: 6 m/s, E + 5 m/s, 300
a – acceleration: 6 m/s2, E + 5 m/s2, 300
F – force: 6 N, E + 5 N, 300
6
+
5
300
you can ONLY add the same
kind (apples + apples)
1. Vectors are drawn to scale in given direction.
2. The second vector is then drawn such that its
tail is positioned at the head of the first vector.
3. The sum of two such vectors is the third vector
which stretches from the tail of the first vector
to the head of the second vector.
This third vector is known as the "resultant" - it is the result of adding the
two vectors. The resultant is the vector sum of the two individual vectors.
So, you can see now that magnitude of the resultant is dependent upon
the direction which the two individual vectors have.
The order in which two or more vectors are added does not effect result.
vectors can be moved around as long as their length
(magnitude) and direction are not changed.
Vectors that have the same magnitude and the same direction
are the same.
Adding A + B + C + D + E yields the
same result as adding C + B + A +
D + E or D + E + A + B + C. The
resultant, shown as the green
vector, has the same magnitude
and direction regardless of the
order in which the five individual
vectors are added.
Example: A man walks 54.5 meters east, then 30 meters west.
Calculate his displacement relative to where he started?

54.5 m, E
+
30 m, W
24.5 m, E

Example: A man walks 54.5 meters east, then again 30 meters east.
Calculate his displacement relative to where he started?
54.5 m, E
30 m, E
+
84.5 m, E

Example: A man walks 54.5 meters east, then 30 meters north.
Calculate his displacement relative to where he started?
54.52 + 30.02 = 62.2
62.2 m, NE
54.5 m, E
30 m, N
+
The sum 54.5 m + 30 m depends on
their directions if they are vectors.
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.
θ = arc tan
62.2 m, NE
30
54.5
30 m, N
q = 290
q
54.5 m, E
So the COMPLETE final answer is :
62.2 m, 290 or 62.2 m @ 290
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.
8.0 m/s, W
v
q
15 m/s, N
v  82  152  17 m/s
8
tan θ   0.5333
15
θ  arctan (0.5333)  [tan 1 (0.5333)]  28.1
The Final Answer :
v  17 m/s @ 28.1 W of N
v  17 m/s, 118.1

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.
R = 142 +232 = 26.93m
12 m, W
tanθ =
6 m, S
20 m, N
q
35 m, E
14
= 0.6087
23
θ = tan-1 (0.6087) = 31.3o
R
14 m
23 m
The Final Answer:
𝑅 = 27 𝑚 @ 310 = 27 𝑚, 310
Vector addition – comparison between
“head-to-tail” and “parallelogram” method
Two methods for vector addition are equivalent.
"head-to-tail" method
of vector addition
parallelogram method
of vector addition
"head-to-tail" method of vector addition
C
B
B
B
+
A
A
parallelogram method of vector addition
B
C
B
+
A
A
The resultant vector 𝐶
is the vector sum of the
two individual vectors.
𝐶=𝐴+𝐵
The only difference is that it is much easier to use "head-to-tail" method
when you have to add several vectors.
What a mess if you try to do it using parallelogram method.
At least for me!!!!
Remember the plane with velocities not at right angles to
each other. You can find resultant velocity graphically,
but now you CANNOT use Pythagorean theorem to get
speed. If you drew scaled diagram you can simply use
ruler and protractor to find both speed and angle. Or you
can use analytical way of adding them. LATER!!!
v1 + v 2 = v
v2
v1
v
 SUBTRACTION is adding opposite vector.
 
C = A - B = A + -B
I WANT YOU TO DO IT NOW
Components of Vectors
– Any vector can be “resolved” into two component vectors.
Vertical component
y – component of the vector
These two vectors are called components.
Ax = A cos q
Ay = A sin q
Ay
A
θ = arc tan
q
Ax
Horizontal component
x – component of the vector
Ay
Ax
if the vector is in
the first
quandrant;
if not, find q from
the picture.
Vector addition: Sum of two vectors gives resultant vector.
A = Ax + Ay
Examle: A plane moves with velocity of 34 m/s @ 48°.
Calculate the plane's horizontal and vertical velocity components.
A m/s @ 48°.
We could have asked: the plane moves with velocity of 34
It is heading north, but the wind is blowing east.
Find the speed of both, plane and wind.
vy
v
v = 34 m/s @ 48° . Find vx and vy
q
vx
vx = 34 m/s cos 48° = 23 m/s
wind
vy = 34 m/s sin 48° = 25 m/s
plane
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.
𝑣𝑥 = 63.5 cos(−320 ) = 53.9 𝑚/𝑠
vx = ?
– 320
Vy
63.5 m/s
𝑣𝑦 = 63.5 sin(−320 ) = −33.6 𝑚/𝑠
Vy = ?
A
If you know x- and y- components of a vector
you can find
the magnitude and direction of that vector:
Let:
Fx = 4 N
Fy = 3 N .
Find magnitude (always positive) and direction.
Fy
F
F= 42 +3 2 =5N
q
q = arc tan (¾) = 370
Fx
F  5N @37 0
Vector addition – analytically
C  AB
Cy
By
C
B
A
Ax
Ay
Cx
Bx

Cx  Ax  Bx  Acos q1  B cos q2
Cy  Ay  By  Asin q1  B sin q2
F = 68 N@ 24°
example:
1
F2
= 32 N @ 65°
F  F1  F2
F
F2
F2
F
1
Fx = F1x + F2x = 68 cos240 + 32 cos650 = 75.6 N
Fy = F1y + F2y = 68 sin240 + 32 sin650 = 56.7 N
F  Fx2  Fy2  94.5 N
q = arc tan (56.7/75.6) = 36.90
F  94.5N @ 370
vector
components
and sum
simple
addition
additions