LESSON PLAN

A. Identity

Unit of Education : Senior High School

Grades : X

Semester : I

Program : -

Subject : Physics

Topic : Physical quantities and units

Sub Topic : Vector

Time Allocation : 5 x 45 minutes (3 meetings)

Standard Competency :

1. Applying the concepts of physical quantities and its

measurement

Basic Competency :

1.2 Adding two or more vectors

Indicators :

1. Students are able to distinguish between scalar quantities and vector quantities

in physics

2. Students are able to draw the vector both in Cartesian and polar coordinate

3. Students are able to determine the resultant of two or more vectors

geometrically

4. Students are able to determine the resultant of two vectors analytically

5. Students are able to determine the dot product and the cross product of two

vectors

B. Learning Objectives

1. Discussing scalar and vector quantities in physics

2. Drawing vector geometrically

3. Drawing vector both in Cartesian coordinate and polar coordinate

4. Determining the resultant of two vector or more geometrically

5. Determining the resultant of two vector analytically

6. Calculating the dot product of two vectors

7. Calculating the cross product of two vectors

C. Learning Materials

Fundamental Concept of Vector

A.1 Vectors

A.1.1 Introduction

Certain physical quantities such as mass or the absolute temperature at some point

only have magnitude. These quantities can be represented by numbers alone, with the

appropriate units, and they are called scalars. There are, however, other physical quantities

which have both magnitude and direction; the magnitude can stretch or shrink, and the

direction can reverse. These quantities can be added in such a way that takes into account

both direction and magnitude. Force is an example of a quantity that acts in a certain

direction with some magnitude that we measure in Newton. When two forces act on an

object, the sum of the forces depends on both the direction and magnitude of the two forces.

Position, displacement, velocity, acceleration, force, momentum and torque are all physical

quantities that can be represented mathematically by vectors. We shall begin by defining

precisely what we mean by a vector.

A.1.2 Properties of a Vector

A vector is a quantity that has both direction and magnitude. Let a vector be denoted

by the symbol . The magnitude of is . We can represent vectors as geometric

objects using arrows. The length of the arrow corresponds to the magnitude of the vector.

The arrow points in the direction of the vector (Figure A.1.1).

Figure A.1.1 Vector as arrow

AA

There are two defining operations for vectors:

(1) Vector Addition: Vectors can be added.

Let and be two vectors. We define a new vector , the “vector addition” of

and , by a geometric construction. Draw the arrow that represents . Place the tail of

the arrow that represents at the tip of the arrow for as shown in Figure A.1.2(a).

The arrow that starts at the tail of and goes to the tip of is defined to be the “vector

addition”. There is an equivalent construction for the law of vector addition. The vectors

and can be drawn with their tails at the same point. The two vectors form the sides

of a parallelogram. The diagonal of the parallelogram corresponds to the vector

, as shown in Figure A.1.2(b).

Vector addition satisfies the following four properties:

(i) Commutivity: The order of adding vectors does not matter

Our geometric definition for vector addition satisfies the commutivity property (i) since

in the parallelogram representation for the addition of vectors, it doesn’t matter which

side you start with as seen in Figure A.1.3.

Figure A.1.2 Geometric sum of vectors

Figure A.1.3 Commutative property of vectors addition

Figure A.1.4 Associative law

(ii) Associativity: when adding three vectors, it doesn’t matter which two we start with

In Figure A.1.4(a), we add , while in Figure A.1.4(b) we add . We

arrive at the same vector sum in either case.

(iii) Identity element for vector addition: there is a unique vector, , that acts as an

identity element for vector addition. This means that for all vectors :

(iv) Inverse element for vector addition: For every vector , there is a unique inverse

vector:

This means that the vector has same magnitude as vector , , but they

point in opposite directions (Figure A.1.5)

(2) Scalars Multiplication of Vectors: vectors can be multiplied by real numbers.

Let be a vector. Let c be a real positive number. Then the multiplication of by c is a

new vector which we denote by the symbol . The magnitude of is c times the

magnitude of (Figure A.1.6a),

Since , the direction of is the same as the direction of . However, the direction

of is opposite of (Figure A.1.6b).

(i) Assosiative Law for Scalar Multiplication: the order of multiplying numbers

is doesn’t matter. Let and be real numbers. Then,

(ii) Distributive Law for Vector Addition: vector addition satisfies a distributive

law for multiplication by a number. Let c be a real number. Then

Figure A.1.7 illustrates this property.

Figure A.1.5 Additive inverse

A

A

Figure A.1.6 Multiplication of vector by (a) , and (b)

A

Ac Ac

Figure A.1.7 Distributive Law for vector addition

)BA(c

BA

A

B

Ac

Bc

Bc

Ac

(iii) Distributive Law for Scalar Addition: The multiplication operation also

satisfies a distributive law for the addition of numbers.

Let b and c be real numbers. Then,

Our geometric definition of vector addition satisfies this condition as seen in Figure

A.1.8.

(iv) Identity Element for Scalar Multiplication: The number 1 acts as an identity

element for multiplication.

A.1.3 Application of Vectors

When we apply vectors to physical quantities it’s nice to keep in the back of our

minds all these formal properties. However from the physicist’s point of view, we are

interested in representing physical quantities such as displacement, velocity,

acceleration, force, impulse, momentum, torque, and angular momentum as vectors. We

can’t add force to velocity or subtract momentum from torque. We must always

Figure A.1.8 Distributive Law for scalar addition

Ac

A Ab

A)cb(

understand the physical context for the vector quantity. Thus, instead of approaching

vectors as formal mathematical objects we shall instead consider the following essential

properties that enable us to represent physical quantities as vectors.

1) Vectors can exist at any point P in space.

2) Vectors have direction and magnitude.

3) Vector Equality: Any two vectors that have the same direction and magnitude are

equal no matter where in space they are located.

4) Vector Decomposition: Choose a coordinate system with an origin and axes. We can

decompose a vector into component vectors along each coordinate axis.

In Figure A.1.9 we choose Cartesian coordinates for the x-y plane (we ignore the -

direction for simplicity but we can extend our results when we need to). A vector at

P can be decomposed into the vector sum,

where is the x-component vector pointing in the positive or negative x-direction, and

is the y-component vector pointing in the positive or negative y-direction (Figure

A.1.9).

5) Unit vectors: The idea of multiplication by real numbers allows us to define a set of

unit vectors at each point in space. We associate to each point P in space, a set of

three unit vectors ( , , ). A unit vector means that the magnitude is one: ,

, and . We assign the direction of to point in the direction of the increasing

x-coordinate at the point P. We call the unit vector at pointing in the +x-direction. Unit

vector and can be defined in a similar manner (Figure A.1.10).

A

yA

x

y

xA

Figure A.1.9 Vector decomposition

AyA

x

y

xA

Figure A.1.10 Choice of unit vector in Cartesian coordinate

zzA

6) Vector components: Once we have defined unit vectors, we can then define the x-

component and y-component of a vector. Recall our vector decomposition, .

We can write the x-component vector, as:

In this expression the term , (without the arrow above) is called the x-component

of vector . The x-component can be positive, zero, or negative. It is not the

magnitude which is given by . Note the difference between the x-

component, , and the x-component vector, . In a similar fashion we define the

y-component, , and the z-component, , of the vector .

A vector can be represented by its three components . We can also

write the vector as:

7) Magnitude: In Figure A.1.10, we also show the vector components .

Using the Pythagorean theorem, the magnitude of the is,

8) Direction: let us consider a vector . Since the z-component is zero, the

vector lies in the x-y plane. Let denote the angle that the vector makes in the

counterclockwise direction with the positive x-axis (Figure A.1.12). Then the x-

component and y-component are:

We can now write a vector in the x-y plane as:

Once the components of a vector are known, the tangent of the angle can be

determined by

which yields

Clearly, the direction of the vector depends on the sign of and . For example, if

both and , then , and the vector lies in the first quadrant. If,

however, and , then , and the vector lies in the fourth quadrant.

9) Vector addition: Let and be two vectors in the x-y plane. Let and denote the

angles that the vectors and make (in the counterclockwise direction) with the

positive x-axis. Then,

In Figure A.1.13, the vector addition is shown. Let denote the angle that

the vector makes with the positive x-axis.

A

yA

x

y

xAFigure A.1.12 Components of a vector in the x-y plane

A

yAx

y

xA

Figure A.1.12 Vector addition with components

A

BxB

yBB

BAC

C

Then the components of are

,

In term of magnitudes and angles, we have

We can write the vector as

A.2 Dot Product

A.2.1 Introduction

We shall now introduce a new vector operation, called the “dot product” or “scalar

product” that takes any two vectors and generates a scalar quantity (a number). We

shall she that the physical quantity of work can be mathematically described by the dot

product between the force and the displacement vectors.

Let and be two vectors. Since any two non-collinear vectors form a plane, we define

the angle to be the angle between vectors and as shown in Figure A.2.1. Note that

can vary from 0 to .

A.2.2 Definition

A

B

Figure A.2.1 Dot product geometry

The dot product of the vectors and is defined to be product of the magnitude of

the vectors and with the cosine of the angle between the two vectors:

Where and represent the magnitude of and respectively. The dot

product can be positive, zero, or negative, depending on the value of cos . The dot

product is always a scalar quantity.

We can give a geometric interpretation to the dot product by writing the definition as

In this formulation, the term A cos is the projection of vector in the direction of

vector . This projection is shown in Figure A.2.2a. So the dot product is the product of

the projection of the length in the direction of with the length of . Note that we

could also write the dot product as

Now the term B cos is the projection of vector in the direction of vector as shown

in Figure A.2.2b. From this perspective, the dot product is the product of the projection

of the length of in the direction of with the length of .

From our definition of our dot product we see that the dot product of two vectors that

are perpendicular to each other is zero since the angle between the vectors is /2 andπ

cos( /2) = 0. π

A.2.3 Properties of Dot Product

A

B

Figure A.2.2a and A.2.2b Projection of vectors and the dot product

cosB

A

B

cosA

The first property involves the dot product between a vector where c is a scalar and

a vector ,

The second involves the dot product between the sum of two vectors and with a

vector ,

Since the dot product is a commutative operation,

The similar definition holds,

A.2.4 Vector Decomposition and the Dot Product

With these properties in mind we can now develop an algebraic expression for the dot

product in terms of components. Let’s choose a Cartesian coordinate system with the

vector pointing along the positive x-axis with positive x-component , i.e.,

The vector can be written as,

We first calculate that the dot product of the unit vector with itself is unity:

Since the unit vector has the magnitude and cos (0) = 1. We note that the same

role applies for the unit vectors in the y and z directions:

The dot product of the unit vector with the unit vector is zero because the two unit

vectors in the y and z-direction:

Similarly, the dot product of the unit vector with the unit vector , and the unit vector

with the unit vector are also zero:

Based on those explanation, for two vectors and ,

the dot product of them now becomes:

A.3 Cross Product

We shall now introduce our second vector operation, called the “cross product”

that takes any two vectors and generates a new vector. The cross product is a type of

“multiplication” law that turns our vector space (law for addition of vectors) into vector

algebra (laws for addition and multiplication of vectors). The first application of the

cross product will be the physical concept of torque about a point P which can be

described mathematically by the cross product of a vector from P to where the force

acts, and the force vector.

A.3.1 Definition

Let and be two vectors. Since any two vectors form a plane, we define the angle θ

to be the angle between the vectors and as shown in Figure A.3.2.1. The magnitude

of the cross product of the vectors and is defined to be product of the

magnitude of the vectors and with the sine of the angle between the two vectors, θ

where A and B denote the magnitudes of and , respectively. The angle between theθ

vectors is limited to the values 0 ≤ ≤ insuring that sin ≥ 0. θ π θ

The direction of the cross product is defined as follows. The vectors and form a

plane. Consider the direction perpendicular to this plane. There are two possibilities, as

shown in Figure A.3.1. We shall choose one of these two for the direction of the cross

product using a convention that is commonly called the “right-hand rule”.

A.3.2 Right-hand Rule for the Direction of Cross Product

Figure A.3.1 Cross product geometry

The first step is to redraw the vectors and so that their tails are touching. Then draw an

arc starting from the vector and finishing on the vector . Curl our right fingers the same

way as the arc. Our right thumb points in the direction of the cross product (Figure

A.3.2).

We should remember that the direction of the cross product is perpendicular to

the plane formed by and . We can give a geometric interpretation to the magnitude

of the cross product by writing the definition as,

The vectors and form a parallelogram. The area of the parallelogram equals the

height times the base, which is the magnitude of the cross product. In Figure A.3.3, two

different representations of the height and base of a parallelogram are illustrated. As

depicted in Figure A.3.3(a), the term is the projection of the vector in the

direction perpendicular to the vector . We could also write the magnitude of the cross

product as,

Now the term is the projection of the vector in the direction perpendicular to

the vector as shown in Figure A.3.3(b).

Figure A.3.2 Right-hand Rule

A

B

Figure A.3.3 Projection of vectors and the cross product

sinB

A

B

sinA

The cross product of two vectors that are parallel (or anti-parallel) to each other is zero

since the angle between the vectors is 0 (or ) and π sin (0) = 0 (or sin ( ) = 0).π

Geometrically, two parallel vectors do not have any component perpendicular to their

common direction.

A.3.3 Properties of the Cross Product

(1) The cross product is anti-commutative since changing the order of the vectors cross

product changes the direction of the cross product vector by the right hand rule:

(2) The cross product between a vector where c is a scalar and a vector is

Similarly,

(3) The cross product between the sum of two vectors and with a vector is

Similarly,

A.3.4 Vector Decomposition and Cross Product

We first calculate that the magnitude of cross product of the unit vector with :

since the unit vector has magnitude and sin ( /2) = 1. By the right hand rule,π

the direction of is in the as shown in Figure A.3.4. Thus .

Similarly we get,

i

j

Figure A.3.4 Cross product of equals to

j

i

k

k

Listing those results to a table,

0 - -

- 0

- 0

Thus, for two vectors and , the cross product of

them now becomes:

The method above is a complicated mathematic. There is another method which is used

to simply this calculation by using determinant.

So, we get,

D. Learning Approach and Method

The learning approach which is used in this activity is student centered and the

learning method which is used in this activity is direct instruction.

E. Learning Activity

1 st meeting (2 x 45 minutes)

1. Pre Activity (±5 minutes)

a. Greeting the student and checking the student attendance

2. Whilst Activity (±80 minutes)

Learning Steps Teacher activities Students activities

Confirmation

Communicating

the indicators and

motivating the

Explaining the indicators

Motivating the students by

asking them

Paying attention to the

explanation and asking the

teacher what they don’t

students

(10 minutes)

Examples:

- Can you mention some

physical quantities which

has both magnitude and

scalar

- How to say your

position now?

understand

Exploration

Explaining the

material

(50 minutes)

Explaining how to draw

vector and determine the

resultant of two vectors

geometrically

Asking some questions to

motivate the student

Paying attention to the

explanation and asking what

they don’t understand

Elaboration

Disccusing some

problems

(30 minutes)

Asking the student to make

some groups

Demonstrating the

application of vector

adding using two spring

balance hang on a block of

mass and ask the student to

find the resultant

geometrically

Making group consists of 3-4

person

Discussing how to find the

resultant of the two spring

balance

Communicating their answer

3. Post Activity (±5 minutes)

a. Teacher facilitates the students to conclude what the have learnt

b. Teacher evaluates the class and allow some students to ask what they still don’t

understand

c. Teacher tells what will be learnt on the next meeting and gives homework

d. Closing the activity

2 nd meeting (1 x 45 minutes)

1. Post Activity (±2 minutes)

b. Greeting and checking the student attendance

2. Whilst Activity

Teacher Activities Student Activities Time allotment

a. Describing the indicators

b. Introducing the student how

to determine the resultant of

two vectors analytically

a. Remembering their

knowledge before

10 minutes

c. Giving each student a

worksheet

b. Solving the problems in

worksheet

15 minutes

d. Discussing the most difficult

problems in worksheet and

allow the student to asking

what they don’t understand

c. Paying attention in

discussion and asking some

question to the teacher

10 minutes

e. Giving the student quiz to

evaluate their understanding

d. Answer the quiz 5 minutes

3. Post Activity (±3 minutes)

a. Teacher facilitates the students to conclude what the have learnt

b. Teacher evaluates the class and allow some students to ask what they still don’t

understand

c. Closing the activity

3 rd meeting (2 x 45 minutes)

1. Post Activity (±5 minutes)

c. Greeting and checking the student attendance

2. Whilst Activity

Teacher Activities Student Activities Time allotment

a. Explaining the indicators

b. Evaluating the quiz done by

the students last meeting

a. Paying attention to the

students

10 minutes

c. Discussing the dot product

of two vectors and giving the

b. Solving the problems 30 minutes

student exercise individually

d. Discussing the cross product

of two vectors and giving

some example to solve

c. Paying attention in

discussion and solving the

problems

30 minutes

e. Giving the student quiz d. Answer the quiz 5 minutes

3. Post Activity (±10 minutes)

d. Teacher facilitates the students to conclude what the have learnt

e. Teacher evaluates the class and allow some students to ask what they still don’t

understand

f. Closing the activity

F. Learning Resources

1. Sunardi, & Irawan, E. I. 2007. Fisika Bilingual. Bandung: Yrama Widya

2. Marthen Kanginan ( 2007). Fisika. Jakarta: Erlangga

3. Chasanah, Chuswatun. 2008. Kreatif Fisika Xa. Klaten: Viva Pakarindo

4. Haliday, Resnick, and Walker. (2001). Fundamental of Physics. Sixth Edition.

New York: John Wiley & Sons, Inc. (Recommended)

5. Presentation media

6. Spring balance and mass

7. Blog (http:/www.pplreal.wordpress.com)

G. Assessment

Cognitive : Worksheet

Affective : Observation sheet