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Vectors and Scalars
 A SCALAR is ANY quantity in physics that
is fully described by MAGNITUDE
 A VECTOR is ANY quantity in physics that
has BOTH MAGNITUDE and DIRECTION.
Magnitude – A numerical value with units.
Scalar Example
Speed
Distance
Age
Heat
Number of horses
behind the school
Vector Example
Magnitude
20 m/s
10 m
15 years
1000
calories
Velocity
Acceleration
Force
Magnitude
& Direction
20 m/s, N
10 m/s/s, E
5 N, West
I guess: 12
A picture is worth a thousand word, at least they say so.
Vectors are typically illustrated by drawing an ARROW above the symbol.
v The arrow is used to convey direction and magnitude.
length = magnitude
6 cm
0
25 above x-axis = direction
displacement x = 6 cm, 250
 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.
 Example of a vector – Velocity of a plane - A 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)
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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.
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?
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 magnitude of resultant velocity (speed v) can be found using Pythagorean theorem
v= v12 +v 22 = (200km/h)2 + (80km/h)2 = 46400km2 /h2
v = 215 km/h
So relative to the ground, the plane moves 215 km/h , SE.
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.
 Vector Addition: 6 + 5 = ?
Till now you naively thought that
6 + 5 = 11.
Not so fast! In vector algebra
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 and 6N force. Adding these two vectors will not necessarily result in a force of 11 N.
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-totail" and “parallelogram” method of vector addition.
We’ll do it, but before that, we have to introduce multiplication of vector by scalar.
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 Vectors that have the same magnitude and the same direction are the same.
Vectors can be moved around as long as their length (magnitude) and
direction are not changed. This is the same vector. It doesn’t matter where it is.
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.
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
 Vector addition - head-to-tail method
vectors: 6 units, E + 5 units,300
Opposite vectors
𝐴⃗
− 𝐴⃗
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
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.
v1 + v 2 = v
 Example: A man walks 54.5 meters east, then 30 meters,
west. Calculate his displacement relative to where
he started?
 Example: A man walks 54.5 meters east, then again 30
meters east. Calculate his displacement relative to where
he started?
 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
30
θ = arc tan
54.5
 = 290
62.2 m, 280 or 62.2 m @ 280
The sum 54.5 m + 30 m depends on their
directions if they are vectors.
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 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
tanθ =
14
= 0.6087
23
θ = tan-1 (0.6087) = 31.3o
𝑅⃗⃗ = 27 𝑚 @ 310
 Vector addition – comparison between “head-to-tail” and “parallelogram” method
Two methods for vector addition are equivalent.
"head-to-tail" method
parallelogram method
v1 + v 2 = v
⃗⃗1 + F
⃗⃗2 = F
⃗⃗
F
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.
 SUBTRACTION is adding opposite vector.
 
C = A - B = A + -B
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 Components of Vectors
– Any vector can be “resolved” into two component vectors. These two vectors are called components.
Ax = A cos 
Ay = A sin 
θ = arc tan
A = Ax + Ay
Ay
Ax
if the vector is in the first quadrant;
if not you find it from the picture.
 Example: 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 = 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
 If you know x- and y- components of a vector you can find the magnitude and direction of that vector:
Example: F = 4 N and F = 3 N . Find magnitude (always positive) and direction.
x
y
F= 42 +32 =5N
 = arc tan (¾) = 370
F  5N @37 0
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 Vector
addition analytically (numbers)
x – component of the sum of two vectors is sum of x-components of individual vectors.
y – component of the sum of two vectors is sum of y-components of individual vectors.
C  AB

Cx  Ax  Bx  Acos 1  B cos 2
Cy  Ay  By  Asin 1  B sin 2
example:
F = 68 N @ 24°
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F2 = 32 N @ 65°
Find
F  F1  F2
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
 = arc tan (56.7/75.6) = 36.90
F  94.5N @ 370