01 Vectors

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PHYSICAL QUANTITIES

VECTORS & SCALARS

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PHYSICAL QUANTITIES

Any number that is used to describe a physical

phenomenon quantitatively using a standard measurable

unit(s).

Example :

Length3 m (meters)

Mass80 kg (kilograms)

Time3600 seconds

Weight100 N (Newtons)

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SCALAR QUANTITIES

Quantities that are described by only a single number which is itsMagnitude. Magnitudejust tells how much of the quantity there

is.

Ex. 10 km, 100 km/hrMass, Volume and Time are scalars

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VECTOR QUANTITIES

Quantities that are described by both magnitudeand the

directionin space.

Ex. 10 km to the left, 100 km/hr eastward

Force , Velocity and Acceleration are vectors

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GRAPHICAL REPRESENTATION OF A VECTOR

Tail

Tip / Head

AVector Notation :

Scalar Notation : A

ANGLE / DIRECTION

A

VECTOR QUANTITIES

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VECTOR QUANTITIES

DIRECTION :

Given in terms of :

STANDARD ANGLES :

Degrees ()

Radians (rad)

180= 3.1416 rad

180= rad

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VECTOR QUANTITIES

DIRECTION : Given in terms of NAVIGATIONAL (COMPASS) BEARINGS :

East

North

West

South

Northeast

Northwest

Southwest

Southeast

E

N

W

S

0

0

90

90

180180

270

NENW

270

SW SE

45

45

135

135

225

225

315

315

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DIRECTION :

With Angles measured or starting from the horizontal (East or West) as reference

, North of East , North of West

, South of East , South of West

Ex:

1. 50, South of East

2. 30, North of West3. 40, South of West

4. 80, North of East

E

N

W

S

50

30

40

80

VECTOR QUANTITIES

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DIRECTION :

With Angles measured or starting from the vertical (North or South) as reference

, East of North , West of North

, East of South , West of South

Ex:

1. 50, East of South

2. 30, West of North3. 40, West of South

4. 80, East of North

E

N

W

S

50

30

40

80

VECTOR QUANTITIES

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VECTOR RESOLUTION

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VECTOR RESOLUTION

A process of combining two or more vectors acting at the

same point on an object to determine a single equivalent

vector known as the Resultant vector.

The resultant has the same effect as the multiple vectors

that originally acts on the object. The resultant vector is

also known as the Net vector.

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VECTOR RESOLUTION

Resultant Can be determined in two ways :

2. Analytical Methods

1. Graphical Methods

These involve plotting and drawing the vectors (using a

convenient scale) and directly measuring the resultant

from these vectors.

These involve no scaled drawings. These are purelycomputation that mostly involves trigonometry.

Provides the most accurate value for the resultant.

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VECTOR RESOLUTION

1. Polygon Method

Graphical Methods

AB C

A

BC

R

The resultant is determined by laying the vectors tail to head in series. Once the last vector is

in placed, the resultant is drawn from the tail of the origin vector up to the tip of the last

vector.

BA

C

R

The commutative property applies here, you can start

at any vector and the resultant is always going to be

the same

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VECTOR RESOLUTIONGraphical Methods

2. Parallelogram Method

AB C

A

BC

R

Start with a pair of vectors drawn from the same origin. Make a parallelogram by projection.

The diagonal will be the resultant of the two vectors. If you have more than two given

vectors, pair the earlier resultant with the next given vector, and so on, the very last diagonal

will be the final resultant.

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VECTOR RESOLUTIONGraphical Methods

NEGATIVE VECTORS

To graphically make a vector negative. Just shift the arrow head 180.

The magnitude remains the same

B B

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VECTOR RESOLUTION

1. Sine & Cosine Laws

Analytical Methods

B

A

RA

B

sin A sin B sin = =A B

RSine Law :

Cosine Law :

R2 = A2 B2 2AB cos +

Useful when given two vectors

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VECTOR RESOLUTIONAnalytical Methods

2. Component Method

Components of a Vector can be thought of as the horizontal & vertical

projections of a vector

Useful for two or more vectors

A

A

AX

AY

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2. Component Method

Mathematically the components of a vector are expressed as :


A

AX= A cos

AY= A sinA

AX= A sin

AY= A cos

Case 1 : measured from

horizontal axis

Case 2 : measured from

vertical axis

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2. Component Method : Sign Convention

The usual vector sign convention

follows the Cartesian coordinatesystem.

x component values:

to the right(or East) arepositive

to the left ( or West) are negative

y component values:

going up(or North) arepositive

going down (or South) are negative.

+x

+y

- x

-y

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2. Component Method

X = AX+ BX+ CX + + ZX

Y = AY+ BY+ CY + + ZY

Algebraic Sum of ALL X-components

Algebraic Sum of ALL Y-components

Computing for the Resultant :

R = X2 +Y

2

= tan-1

X

Y

Standard Sign Convention

IfX is + , it is going to the right or east

IfX is , it is going to the left or west

IfY is + , it is going upward or north

IfY is , it is going downward or south


Note : The angle computed here is ALWAYS measured from the

horizontal axis. ALWAYS between zero to 90. Refer to the sign

convention above for the correct bearing

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UNIT VECTORS

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+x

+y

- x

-y

+z

-z

Three-Dimensional Coordinate System+y

-y+x

-x

AX = A cos

AY= A sin

AX = A cosAY= A cos

AZ= A cos

VECTOR COMPONENTS :

Review :

A

A

UNIT VECTORS

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UNIT VECTORS

A unit vector is avector whose magnitude is equal to oneand dimensionless. They are used to specify a determined

direction or simply pointer vectors.

A unit vector is sometimes denoted by replacing the arrow on a vector

with a "^" or just adding a "^" on a boldfaced character .

Unit vector for X-component vector

Unit vector for Y-component vector

Unit vector for Z-component vector

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UNIT VECTORS3D Vector is written in rectangular coordinate system as :

AX = A cos AY = A cos AZ = A cos

Components are :

Magnitude of the 3D Vector :

Note : 3D vector becomes a 2D vector , when ONE of ANY of the components

becomes zero.

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UNIT VECTORS

+z

-z

+y

-y

+x

-x

Plot: Vector A:

A= 3 5+ 4k^

A

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Addition/Subtraction: Simply get the sum or difference between

the same components.

Vector A:

A= 3 5+ 4k

Vector B:

B= 2 2 5k

A+B= 5 7 k

A= 3 5 + 4k

B= 2 2 5k+

AB= 3+ 9k

A= 3 5 + 4k

B= 2 2 5k

^ ^

^

^

^

^

^

^

UNIT VECTORS

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APPLICATION OF VECTOR RESOLUTION :

RELATIVE VELOCITY

Velocity: A vector quantity that is the rate of change in

position (displacement) over a time interval

Speedis the scalar part or the magnitude of velocity

Relative Velocity : The observed velocity of an object withrespect or relative to where the observer is oriented to

(frame of reference of the observer) .

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Double Subscript Notation:

vAB This reads as velocity of objectArelative to object B

Example :Velocity of car on the road (earth)

vCE

RELATIVE VELOCITY

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Double Subscript Notation:

vAC This reads as velocity of objectArelative to object C

Given two objects with different relative velocities :

If we want to know the relative velocity ofAwith

respect to C, then we get the resultant of these two :

vBC

This reads as velocity of object Brelative to object C

vAC= vAB+ vBC

vAB This reads as velocity of objectArelative to object B

RELATIVE VELOCITY

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1. An airplane heading due south with an airspeed of 200kph is in a cross

wind of 10kph due west. How far does the airplane go in 2 hours and in whatdirection?

ANS : S = 400.5 km, = 87.138S of W

RELATIVE VELOCITY

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RELATIVE VELOCITY2. A boat is capable of making 9kph in still water is used to cross a river flowing at a

speed of 4kph.

a) At what angle () should the boat be directed so that the motion will be straight

across the river?

b) What is the resultant speed relative to the shore (earth)?

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DOT & CROSS PRODUCTS

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The dot product is denoted by " " between two vectors. The dot product

of vectors Aand Bresults in a scalarvalue. Dot product is given by the

relation :

The dot product follows the commutative and distributive properties

Where is the angle betweenA&B

DOT PRODUCT

Alternative Equation

(If is not given, but the component are)

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Given two vectors

P = AxB =

(+)(+) (+)

() () ()

A=Ax +Ay+Azk B= Bx+By+Bzk^ ^

CROSS PRODUCTThe cross product is denoted by "x" between two vectors. The cross

product of vectors Aand Bresults in avector.

Cross Product obtained using Determinants (3x3 matrix)

Cross Product obtained using this formula

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CROSS PRODUCTMagnitude of the Cross Product

The cross product has the following properties

Where P(the magnitudeof the cross product) is

equal to the area of the parallelogram formed by the

two vector.

OR

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DOT and CROSS PRODUCT

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APPLICATION OF VECTOR RESOLUTION :

NAVIGATION via Displacement

Displacement(s): A vector quantity that is the change in

position of an object.

Distance is the scalar counter part of displacement. It may

vary because there is a multiple (if not infinite) number of

ways to get from one point to another.

The magnitude of the displacement is considered as

a distance, in fact it is the shortest possible value for

distance.

s This reads as displacement vector S

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NAVIGATION via Displacement1. Ace City lies 30 km directly south of Blues City. A bus, beginning at Ace

City travels 50 km at 37north of east to reach Chapel City. How far and in

what direction must the bus go from Chapel City to reach Blues City?

Blues

Ace

Chapel

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NAVIGATION via Displacement2. An escaped convict runs 1.70 km due east of the prison. He then runs due

north to a friend's house. If the magnitude of the convict's total displacement

vector is 2.50 km, what is the direction of his total displacement vector with

respect to due east?


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3. [P1.24] A sailor in a small sailboat encountered shifting winds. She sails 2km

east, then 3.5km southeast, and then an additional distance in an unknown

direction. Her final position is 5.8km directly east of the starting point. Find the

magnitude and direction of the third leg of the journey.Given :

A = 2km

5.8 kmStart Finish

C = ?45

Required : C &

Solution : Use Component Method

|R| = 5.8 km R = Rx2+ Ry

2

By observation Ry= 0 , Hence R = Rx = 5.8km

Rx = Ax+ Bx+ Cx

Ry = Ay+ By+ Cy

5.8km= +2km + 3.5km(sin 45) + Cx

Vectors : (+) and (+).

5.8km= 4.475km + Cx1.325km=Cx

0= 0 3.5km(cos 45) + Cy2.475km=Cy

u

v

C = Cx2 + Cy

2 = (1.325 km)2+ (2.475 km)2

C = 2.807km



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3. 1.24. A sailor in a small sailboat encountered shifting winds. She sails 2km

east, then 3.5km southeast, and then an additional distance in an unknown

direction. Her final position is 5.8km directly east of the starting point. Find the

magnitude and direction of the third leg of the journey.Given :

A = 2km

5.8 km

Start Finish

C = ?45

Required : C &

Divide by uv

2.475km Cy

1.325km Cx=

1.868 = tan

= tan-1 ( 1.868 )

= 61.837N of E


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