PHYSICAL QUANTITIES VECTORS & SCALARS

PHYSICAL QUANTITIES

Any number that is used to describe a physical phenomenon quantitatively using a standard measurable unit(s).

Example :

Length – 3 m (meters)

Mass – 80 kg (kilograms)

Time – 3600 seconds

Weight – 100 N (Newtons)

SCALAR QUANTITIESQuantities that are described by only a single number which is its Magnitude. Magnitude just tells how much of the quantity there is.

Ex. 10 km, 100 km/hr

Mass, Volume and Time are scalars

VECTOR QUANTITIESQuantities that are described by both magnitude and the direction in space.

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

Force , Velocity and Acceleration are vectors

GRAPHICAL REPRESENTATION OF A VECTOR

Tail

MAGNITUDE Tip / Head

A Vector Notation :

Scalar Notation : A

θ ANGLE / DIRECTION

A

VECTOR QUANTITIES

VECTOR QUANTITIES

DIRECTION :

Given in terms of :

STANDARD ANGLES :

Degrees (°)Radians (rad)

180° = 3.1416 rad

180° = π rad

VECTOR QUANTITIESDIRECTION : Given in terms of NAVIGATIONAL (COMPASS) BEARINGS :

East

North

West

South

Northeast

Northwest

Southwest

Southeast

E

N

W

S

0°

0°

90°

90°

180°180°

270°

NENW

270°

SW SE

45°

45°

135°

135°

225°

225°

315°

315°

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

θ, North of East θ, North of Westθ, South of East θ, South of West

Ex: 1. 50° , South of East2. 30° , North of West3. 40° , South of West4. 80° , North of East

E

N

W

S

50°

30°

40°

80°

VECTOR QUANTITIES

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

θ, East of North θ, West of Northθ, East of South θ, West of South

Ex: 1. 50° , East of South2. 30° , West of North 3. 40° , West of South4. 80° , East of North

E

N

W

S

50°

30°

40°

80°

VECTOR QUANTITIES

VECTOR RESOLUTION

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.

VECTOR RESOLUTIONResultant Can be determined in two ways :

2. Analytical Methods

1. Graphical MethodsThese 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 purely computation that mostly involves trigonometry. Provides the most accurate value for the resultant.

VECTOR RESOLUTION

1. Polygon Method

Graphical Methods

A B C

A

B C

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.

B A

C

R

The commutative property applies here, you can start at any vector and the resultant is always going to be the same

VECTOR RESOLUTIONGraphical Methods

2. Parallelogram Method

A B C

A

B C

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.

VECTOR RESOLUTIONGraphical Methods

NEGATIVE VECTORSTo graphically make a vector negative. Just shift the arrow head 180°. The magnitude remains the same

B −B

VECTOR RESOLUTION

1. Sine & Cosine Laws

Analytical Methods

B

A

RθA

θB

θ

sin θA sin θB sin θ= =A B RSine Law :

Cosine Law :

R2 = A2 B2 2AB cos θ+ −

Useful when given two vectors

VECTOR RESOLUTIONAnalytical Methods

2. Component MethodComponents of a Vector can be thought of as the horizontal & vertical projections of a vector

Useful for two or more vectors

A

θ

A

θ

AX

AY

VECTOR RESOLUTIONAnalytical Methods

2. Component MethodMathematically the components of a vector are expressed as :

Useful for two or more vectors

A

θ

AX = A cosθ

AY = A sinθA

θ

AX = A sinθ

AY = A cosθ

Case 1 : θ measured from horizontal axis

Case 2 : θ measured from vertical axis

VECTOR RESOLUTIONAnalytical Methods

2. Component Method : Sign Convention

The usual vector sign convention follows the Cartesian coordinate system.x – component values :to the right (or East) are positiveto the left ( or West) are negativey – component values :going up (or North) are positivegoing down (or South) are negative.

+x

+y

- x

-y

VECTOR RESOLUTIONAnalytical Methods

2. Component Method

ΣX = AX + BX + CX + ··· + ZX

ΣY = AY + BY + CY + ··· + ZY

Algebraic Sum of ALL X-components

Algebraic Sum of ALL Y-components

Computing for the Resultant :

R = ΣX2 + ΣY

2

θ = tan-1

ΣX

ΣY

Standard Sign Convention

If ΣX is + , it is going to the right or east

If ΣX is − , it is going to the left or west

If ΣY is + , it is going upward or north

If ΣY is − , it is going downward or south

Useful for two or more vectors

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

UNIT VECTORS

+x

+y

- x

-y

+z

-z

Three-Dimensional Coordinate System+y

-y+x

-x

α

β

γ

θ

AX = A cosθ

AY = A sinθ

AX = A cos α AY = A cos β AZ = A cos γ

VECTOR COMPONENTS :

Review :

A

AUNIT VECTORS

UNIT VECTORSA unit vector is a vector whose magnitude is equal to one and 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

UNIT VECTORS3D Vector is written in rectangular coordinate system as :

AX = A cos α AY = A cos β AZ = A cos γ

Components are :

Magnitude of the 3D Vector :

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

UNIT VECTORS

+z

-z

+y

-y

+x

-x

Plot: Vector A :A = 3î − 5ĵ + 4k ^

A

Addition/Subtraction: Simply get the sum or difference between the same components.

Vector A :

A = 3î − 5ĵ+ 4k Vector B :

B = 2î − 2ĵ − 5k

A+B = 5î − 7ĵ − k

A = 3î − 5ĵ + 4k B = 2î − 2ĵ − 5k +

A−B = î − 3ĵ + 9k

A = 3î − 5ĵ + 4k B = 2î − 2ĵ − 5k −

^ ^

^

^

^

^

^

^

UNIT VECTORS

APPLICATION OF VECTOR RESOLUTION : RELATIVE VELOCITY

Velocity : A vector quantity that is the rate of change in position (displacement) over a time interval

Speed is the scalar part or the magnitude of velocity

Relative Velocity : The observed velocity of an object with respect or relative to where the observer is oriented to (frame of reference of the observer) .

Double Subscript Notation :

vAB This reads as velocity of object A relative to object B

Example : Velocity of car on the road (earth)

vCE

RELATIVE VELOCITY

Double Subscript Notation :

vAC This reads as velocity of object A relative to object C

Given two objects with different relative velocities :

If we want to know the relative velocity of A with respect to C , then we get the resultant of these two :

vBC This reads as velocity of object B relative to object C

vAC = vAB + vBC

vAB This reads as velocity of object A relative to object B

RELATIVE VELOCITY

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 what direction?

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

RELATIVE VELOCITY

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 (θ) 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)?

DOT & CROSS PRODUCTS

The dot product is denoted by " ● " between two vectors. The dot product of vectors A and B results in a scalar value. Dot product is given by the relation :

θ

The dot product follows the commutative and distributive properties

Where θ is the angle between A & B

DOT PRODUCT

Alternative Equation

(If θ is not given, but the component are)

Given two vectors

P = AxB =

(+)(+) (+)

(−) (−) (−)

A = Ax î + Ay ĵ + Az k B = Bx î + By ĵ + Bz k ^ ^

CROSS PRODUCTThe cross product is denoted by "x " between two vectors. The cross product of vectors A and B results in a vector.

Cross Product obtained using Determinants (3x3 matrix)

Cross Product obtained using this formula

CROSS PRODUCTMagnitude of the Cross Product

The cross product has the following properties

Where P (the magnitude of the cross product) is equal to the area of the parallelogram formed by the two vector.

OR

DOT and CROSS PRODUCT

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

NAVIGATION via Displacement1. Ace City lies 30 km directly south of Blues City. A bus, beginning at Ace City travels 50 km at 37° north 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

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?

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

B = 3.5km

5.8 kmStart Finish

θ

C = ? 45°

Required : C & θ

Solution : Use Component Method|R| = 5.8 km R = Rx

2 + Ry2

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°) + Cy

2.475km =Cy

C = Cx2 + Cy

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

C = 2.807km

NAVIGATION via Displacement

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

B = 3.5km

5.8 km

Start Finish

θ

C = ? 45°

Required : C & θ

Divide by uv2.475km Cy

1.325km Cx

=

1.868 = tan θ

θ = tan-1 ( 1.868 )

θ = 61.837° N of E

NAVIGATION via Displacement