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Notes for motion in a plane chapter of class 11 physics. Dronstudy provides free comprehensive chapterwise class 11 physics notes with proper images & diagram.

**How to Locate Points in Space?
**Positions in space are designated relative to coordinate systems. The Cartesian coordinate system is a particularly convenient coordinate system in which position are designated by distances (x, y, z) along three perpendicular axes that intersect at a point called the originIn a

A **frame of reference **is another name for the particular coordinate system with respect to which we are making observations of physical phenomena.

A **scalar** quantity requires only a number for its complete description. Mass, volume, density, pressure and temperature are all examples of scalar quantities. The mathematics of scalar quantities is the ordinary algebra of numbers.

**Vector **quantities requireÂ both *magnitude *and *direction *for its complete description. Velocity, acceleration, force and momentum are the examples of vector quantities. A vector can be represented graphically by a directed line segment. The length of the line segment represents the vector's magnitude and its angle with respect to some coordinate system specifies its direction.

We will represent vectors byÂ bold face type letters, with an arrow over the letter such as . When written by hand the same representation may be used.

The magnitude of a vector will be represented by *italic *type letters such as *a*, *b*, *c *etc.

If two vectors have the same direction, they are **parallel**. If they have the same magnitude and the same direction, they are **equal**, no matter where they are located in space.

If two vectors have the same direction, they are **parallel**. If they have the same magnitude and the same direction, they are **equal**, no matter where they are located in space.

The **negative of a vector **is defined as a vector having the same magnitude as the original vector but the opposite direction.

When two vectors **Â **and Â have opposite directions, whether their magnitudes are the same or not, we say that they are *antiparallel*.

**Vector Algebra**

**Addition of Vectors
(i) Geometrical Method
**Two vectors Â Â and Â may be added geometrically by drawing them to a common scale and placing then

**(ii) Analytical MethodÂ ****(Parallelogram law of vector addition)
**If the two vectors Â and are given such that the angle between them is . The magnitude of the resultant vector of their vector addition is given by

and its direction is given by angleÂ Â with

vectorÂ

It is a common error to conclude that if Â the magnitude ofÂ Â should be just equal to the magnitude ofÂ Â plus the magnitude ofÂ Â In general, the conclusion is wrong; one can see that

**Subtraction of Vectors
**To subtractÂ fromÂ , reverse the direction ofÂ to get - ; then add - toÂ

**Resolution of a Vector**

The (*scalar*) components, a_{x} and a_{y} of any two dimensional vectorÂ Â are found by dropping perpendicular lines from the ends ofÂ Â onto

the coordinate axes. The components are given by

andÂ

whereÂ Â is the angle from the positive direction of the x axis to the direction ofÂ Â The algebraic sign of a component indicates its direction along the associated axis.

Given the components, we can reconstruct the vector from

andÂ

*Application 1*

*Find the components of the vectors and as shown in the figure, if a = 2 unitsÂ and b = 3 units.*

*Solution*

units

Â units

Â units

Â units

*Application 2*

*Find the magnitude and direction of the resultant vector Â where Â and are vectors shown in figure.*

*Solution*

Â units

Â units

= 4 units

units

**Note
**There is one slight complication in using equationÂ Â to findÂ SupposeÂ m andÂ m; thenÂ Â But there are two angles having tangents of -1, namely 135Â° and 315Â° (or -45Â°).Â In general, any two angles that differ by 180Â° have the same tangent. To decide which is correct, we have to look at the individual components. Because

**Unit Vectors
**A

In an

The vectorÂ Â can be written in terms of its components as

When two vectorsÂ andÂ Â are represented in terms of their components, we can express the vector sumÂ Â using unit vector as follows

=

In

whereÂ Â is the unit vector along the

*Application** 3*

**IfÂ andÂ **

*(a) Find the components of
(b) Find the magnitude ofÂ Â and its angle with x-axis.*

*Solution*

(a)Â

ThusÂ ;

(b)

**Multiplication of Vectors
(i) Multiplication of a vector by a scalar
**The product of a scalar

**(ii) Multiplication of a vector by another vector
(a) Scalar or Dot Product
**The scalar (or dot) product of two vectorsÂ andÂ Â is writtenÂ Â and is the scalar quantity

In whichÂ Â is the angle between the directions ofÂ andÂ .Â The scalar product may be positive, zero, or negative, depending on the value of . A scalar product is the product of magnitude of one vector and the component of the second vector along the direction of the first vector.

**Some properties of Dot product
**1) It is commutative, i.e.Â =Â

2) It is distributive over addition and subtraction i.e.Â

3) IfÂ andÂ areÂ perpendicularÂ = 0.

4)Â IfÂ andÂ Â are parallel then =a.b

5) Square of a vector is defined asÂ = a

6)Â In

Thus,Â

*Application 4*

*If ; Find *

**Solution**

**(b) Vector or Cross product
**The vector (or cross) product of two vectors and is written as and is a vector whose magnitude c is given by

In which is the smaller of the angles between the direction of and . the direction of is perpendicular to the plane defined by and and is givenÂ by right hand rule.

=

and

where is the unit vector.

The vector Â is directed perpendicular to the plane formed byÂ and Â The direction of vectorÂ Â may be obtained by using the ** Right Hand Thumb Rule**. Stretch all the fingers and thumb of your right hand such that they are perpendicular to each other.Â Now alignÂ your hand such that its plane is perpendicular to the plane formed by vectorsÂ and . Align the stretched fingers along the direction of the vector written

**Properties of Cross Product**

1) It is not commutativeÂ .Â In factÂ

2)Â It is distributive over addition and subtraction

3) IfÂ , thenÂ and ifÂ ,Â

4) IfÂ ,Â ,Â Â be the unit vectors along in the positive directions of *x*, *y*, and *z* axes thenÂ

;Â ;Â Â (maintain cyclic order)

5)Â The cross product may also be expressed by the determinant

where

andÂ

*Application** 5*

*If ; Find *

*Solution*

SinceÂ ;Â

;Â

Good