Vector

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Janak Singh Saud [Kanjirowa National H. S. School]

Vector• Additional Mathematics

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VECTOR

What we learn ?Scalar Product of vectorsScalar Product of vectors and Standard Unit vectorsScalar Product of vectors in Terms of CoordinatesProperties of Scalar Product of Two Vectors

Vector GeometrySection Formula, Mid-point formula and centroid formulaSome Important Theorem, Properties and Relations of Geometric Configuration

Janak Singh Saud [Kanjirow

a National H

. S. School]

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Are You Ready ?

• Vector and its components and representation• Magnitude and direction of vectors• Different types of vectors• Operations of vector• Addition of Vector with triangle, parallelogram and

polygonal laws.• Subtraction of vectors• Scalar Multiplication of vector and parallel vectors

in terms of scalar Multiplication


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Scalar Quantity:Those physical quantities which have

only magnitude but no direction are called scalar quantities of simply scalar. For example length, area, volume, mass, density etc.


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Vector Quantity:

Those physical quantities which have both magnitude and direction are called vector quantities or simply vector. E.g.. Velocity, displacement, acceleration, force, pressure. ….


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Vector


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Vector in coordinate form

Here , 3 is called X- component and 2 is called Y- component

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anjirowa N

ational H. S. School]


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Vector in graph

1 2 3 4 5 6-1-2-3-4-5-6

x

1

2

3

4

5

6

-1

-2

-3

-4

-5

-6

y

A(3, 2)

O


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Magnitude(Modulus) of a vector

The length between the initial and terminal point of a vector is known as magnitude of a vector.


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Cont……Janak Singh Saud [K

anjirowa N



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Magnitude Of vector

1 2 3 4 5 6-1-2-3-4-5-6

x

1

2

3

4

5

6

-1

-2

-3

-4

-5

-6

y

A(3, -2)

O

3

- 2

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Direction of a vector:

The direction of a vector is the angle made by a vector with the positive X-axis in anticlockwise direction


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TYPES OF VECTORS:

1. Position vector2. Row Vector3. Column Vector4. Null or Zero Vector5. Unit Vector6. Equal Vectors7. Negative of a vector8. Parallel Vectors


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1. POSITION VECTOR:

A vector whose initial point is at origin is known as position

vector.


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Position Vector Of A(3, -2)

1 2 3 4 5 6-1-2-3-4-5-6

x

1

2

3

4

5

6

-1

-2

-3

-4

-5

-6

y

O

A(5, 4)


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2. Row Vector:

A vector whose X-component and Y-component are written in row form is known as row vector.

Row vector = (X-component , Y-component)


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3. COLUMN VECTOR:

A vector whose X-component and Y-component are written in column form is known as column vector


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4. Null Vector or Zero Vector

A vector whose magnitude is zero is called null or zero vector.


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5. Unit Vector

A vector whose magnitude is 1 is known as unit vector.


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Unit Vectors Along X-axis and Y-axisJanak Singh Saud [K

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1 2 3 4 5 6-1-2-3-4-5-6

x

1

2

3

4

5

6

-1

-2

-3

-4

-5

-6

y

(1,0)(0, 1)


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Cont. ….

Any vector can be changed into unit vector by dividing that vector by its magnitude


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Cont…….Janak Singh Saud [K

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6. Equal vectors:

Two vectors are said to be equal if and only if they have same magnitude and same direction


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7. Negative of a vector:Janak Singh Saud [K

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Vector Negative Vector-


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8. Parallel Vectors

Two vectors and are said to be parallel if = k or = m , where k and m are positive scalar and negative scalar respectively.

= = =


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Vector addition and subtraction

+

=

= =

+


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Vector Addition ExampleJanak Singh Saud [K

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A

Triangle Law of Vector Addition:

BC

It states that when all sides of a triangle represent vectors, then the sum of any two sides of the triangle taken in the same order is given by the remaining side of the triangle in the opposite order.


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Cont……..

A

BC


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A B

O


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Parallelogram Law of Vector addition

If two adjacent sides of a parallelogram through a point represents two vectors in magnitude and direction, then their sum is given by the diagonal of the parallelogram through the same point in magnitude and direction

A B

DC


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A B

DC

Parallelogram Law of Vector additionJanak Singh Saud [K

anjirowa N


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Example

Soln . Here, x = 2 and y = 3


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Scalar or Dot product of two vectors

B

A

O


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Example

Solution :

By definition,

= 0 (scalar)


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Soln . By definition,

6 Cos 450

= 12 (scalar)


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Next way :If angle is not given

= - 12 + 12 = 0

Now,


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Case I [Perpendicular/Orthogonal]

By definition

= 0

Thus when two vectors are perpendicular to each other, then their scalar product is zero

Conversely,

Thus, if the scalar product of two vectors is zero, then the vectors are perpendicular (orthogonal) to each other.

B

A

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Example:

= - 12 + 12 = 0

Now,


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Case II[Like and parallel]

When two vectors are like and parallel then angle between them is 00

By definition,


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EXAMPLE

Soln. Here,

= . = + = 8 + 18 26=

a = = = =

b = = = = =

= = 132 = 26

=


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Case III [Unlike and parallel Vectors

When two vectors are unlike and parallel then angle between them is 1800

By Definition

B A

1800

O


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EXAMPLE

Soln : =

and =

= = =

= = = = 82

= . = - 4 8 + 5 (-10) = -82

Therefore given vectors are parallel but unlike.


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Scalar product in terms of components

Let A(x1, y1) and B(x2, y2) be two points then their position vectors are defined by

and

= x1 x2 (1) + x1 y2 (0) + y1 x2 (0) + y1 y2 (1)


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Magnitude of Unit vectors Janak Singh Saud [K

anjirowa N


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Now


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NEXT METHODJanak Singh Saud [K

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Angle Between Two vectors

B(x2,y2)

A(x1, y1)

O


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Properties of scalar ProductThe scalar product of two vectors holds the following properties:

is a scalar or constant quantity1.

2.

4.

3.

5.

6.

7.

8.


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Proof:and

Then.=

= + + -

= - 2 + 0 + 0 - 15

Thus, Dot product of two vectors is always scalar.= -17 ( scalar)

(1)Janak Singh Saud [K

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(2)

Proof:and

Then.

=

+

= - 17

=

=


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Vector Geometry is in next slide

Thank You


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Education

Vector