Vectors have both magnitude and direction, while scalars only have magnitude. There are two main methods for adding vectors graphically: the head-to-tail method and the parallelogram method. Vectors can also be represented and added using their horizontal and vertical components. The dot product of two vectors yields a scalar that indicates the cosine of the angle between the vectors, while the vector product yields a vector that is perpendicular to both original vectors and whose magnitude depends on the angle between them.

1. Vectors and Scalars

3. 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
Number of
horses
behind the
school
1000
calories
I guess: 12

4. Cartesian coordinates
A Cartesian coordinate system is defined as a set of two or more axes
with angles of 90° between each pair. These axes are said to be
orthogonal to each other

5. A VECTOR is ANY quantity in
physics that has BOTH
MAGNITUDE and DIRECTION.
Vector Magnitude &
Direction
Velocity 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.
Tail
Head
L
H
250
length = magnitude
6 cm
250 above x-axis = direction
displacement x = 6 cm, 250
Vectors are typically illustrated by drawing an ARROW above the symbol.
The arrow is used to convey direction and magnitude.
v

6. 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.

7. Cartesian Representation of Vectors
𝑃 = (-2,-3)
Q = (3, 1)
R = (-3, -
1)
S = (2, 3).
For simplicity, we can shift the beginning of
a vector to the origin of the coordinate
system

8. we see that we can represent a vector in Cartesian
coordinates as
Called components

9. A resultant (the real one) velocity is sometimes
the result of combining two or more velocities.

10. 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.
resulting velocity relative
to the ground is 280 km/h
2. It’s Texas: the wind changes
direction suddenly 1800.
Velocity vectors are now in
opposite direction.
Flying against a 80 km/h wind, the
plane travels only 120 km in one
hour relative to the ground.
e
e
200km
h
80
200km
h
km
h
200km
h
80 km
h
280km
h
120km
h

11. 3. The plane encounters a crosswind of 80 km/h.
Will the crosswind speed up the plane, slow it down, or have no effect?
To find that out we have to add these two vectors.
The sum of these two vectors is called RESULTANT.
200km
h
80 km
h
RESULTANT
RESULTANT VECTOR
(RESULTANT VELOCITY)
The magnitude of resultant velocity (speed v)
can be found using Pythagorean theorem
v = 215 km/h
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.
So relative to the ground, the plane
moves 215 km/h southeasterly.
2 2 2 2 2 2
1 2
v= v +v = (200km/h) + (80km/h) = 46400km /h
80km
h
200km
h

12. To find the direction
θ = tan−1
𝑣𝑦
𝑣𝑥
=
200
80

13. Not so fast
Vector Addition: 6 + 5 = ?
Till now you naively thought that 6 + 5 = 11.
In vector algebra
6 + 5 can be 10 and 2, and 8, and…
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.

14. 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.
1. Graphical Vector Addition and
Subtraction

15. 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.

16. Multiplying vector by a scalar
Multiplying a vector by a
scalar will ONLY CHANGE
its magnitude.
Opposite vectors
One exception:
Multiplying a vector by “-1” does not
change the magnitude, but it does
reverse it's direction
Multiplying vector by 2 increases its magnitude
by a factor 2, but does not change its direction.
A 2A 3A ½ A
A
- A
– A
– 3A

18. Adding vectors graphically

19. 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.
Vector addition - head-to-tail method
6
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
+
5
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.
300
you can ONLY add the same
kind (apples + apples)

20. 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.
The order in which two or more vectors are added does not effect result.
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.

21. Two methods for vector addition are equivalent.
"head-to-tail" method
of vector addition
parallelogram method
of vector addition
Parallelogram method
Vector addition – comparison between
“head-to-tail” and “parallelogram” method

22. "head-to-tail" method of vector addition
parallelogram method of vector addition
The resultant vector 𝐶
is the vector sum of the
two individual vectors.
𝐶 = 𝐴 + 𝐵
C
+
C
+
A
B B
A
B
A
B
A
B

23. 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!!!!

24. Adding vectors by components

25. Components of Vectors
– Any vector can be “resolved” into two component vectors.
These two vectors are called components.
Horizontal component
x – component of the vector
Vertical
component
y
–
component
of
the
vector
Vector addition: Sum of two vectors gives resultant vector.
Ax = A cos 
Ay = A sin 

A
Ax
Ay
x y
A = A + A
θ = tan−1
Ay
Ax
if the vector is in
the first
quandrant;
if not, find  from
the picture.

26. Unit vectors
There is a set of special vectors that make much of the math
associated with vectors easier Called unit vectors,
they are vectors of magnitude 1 directed along the main
coordinate axes of the coordinate system.

27. A
v = 34 m/s @ 48° . Find vx and vy
vx = 34 m/s cos 48° = 23 m/s wind
vy = 34 m/s sin 48° = 25 m/s plane
vx
vy

v
Examle: A plane moves with velocity of 34 m/s @ 48°.
Calculate the plane's horizontal and vertical velocity components.
We could have asked: the plane moves with velocity of 34 m/s @ 48°.
It is heading north, but the wind is blowing east.
Find the speed of both, plane and wind.
𝑉 = 23𝑥 + 25 𝑦

28. 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
320
vx = ?
Vy = ?
Vy
𝑣𝑥 = 63.5 cos(3280) = 53.9 𝑚/𝑠
𝑣𝑦 = 63.5 sin(328) = −33.6 𝑚/𝑠
𝑉 = 53.9 𝑥 − 33.6𝑦

29. If you know x- and y- components of a vector you can find
the magnitude and direction of that vector:
A
Let:
Fx = 4 N
Fy = 3 N .
Find magnitude (always positive) and direction.
2 2
F= 4 +3 =5N
 = tan−1
(¾) = 370
 0
F 5N @37
Fx
Fy

F

30. C
A
B
Ay
Ax
By
Bx
Cx
Cy
Vector Addition Using
Components
x x x 1 2
y y y 1 2
C A B
C A B Acos Bcos
C A B Asin Bsin
 
 
 

   
   

31. 1
F
F
2
F
2
F
example:
1
F = 68 N@ 24° = 32 N @ 65°
2
F
2
1
F
F
F 

Fx = F1x + F2x = 68 cos240 + 32 cos650 = 75.6 N
Fy = F1y + F2y = 68 sin240 + 32 sin650 = 56.7 N
N
5
.
94
F
F
F 2
y
2
x


  = tan−1(
56.7
75.6
)= 36.90
N
F . @
 0
945 37

32. Exercise:

34. Vector and dot product
what is dot product?
The dot product of two vectors A and B is defined
as the scalar value AB cos α
the scalar product is often referred
to as the dot product.

35. • If two vectors form a 90° angle, then the scalar product has the value zero
𝐴. 𝐵 = 𝐵. 𝐴
To find the angle between two vectors use

36. Example:
Two vectors A=(3,2,4)N and B=(1,2,3)N , find the magnitude of A.B and the
angle between them
Answer:
𝐴. 𝐵 = 3 ∗ 1 + 2 ∗ 2 + 3 ∗ 4 = 19N2
𝐴 = 32 + 22 + 42 = 5.4𝑁
𝐵 = 12 + 22 + 32 = 3.7𝑁
cos−1
19
5.4 ∗ 3.7
= 18

37. notes
• Scalar product of unit vecors as follow
• For the scalar product, the same distributive property that is valid for the
conventional multiplication of numbers holds:

38. Vector product

39. Easier method
The angular relationship of two vectors A and B is

40. Notes:
• It is important to realize that for the vector product, the order of the factors
matters
• the vector product with itself is always zero:
• If we have three vectors and we want to find the vector product between them

41. Vector product of unit vectors

42. Example:
c. Find the magnitude of AxB