Chapter-3

- Vectors

Chapter 3 vectors

- In physics we have Phys. quantities that can be

completely described by a number and are known as

scalars. Temperature and mass are good examples

of scalars. - Other physical quantities require additional

information about direction and are known as

vectors. Examples of vectors are displacement,

velocity, and acceleration. - In this chapter we learn the basic mathematical

language to describe vectors. In particular we

will learn the following -

Geometric vector addition and subtraction

Resolving a

vector into its components

The

notation of a unit vector

Addition and subtraction vectors by components

Multiplication of a vector by a scalar

The scalar (dot) product of two vectors

The vector (cross) product of two vectors

Ch 3-2 Vectors and Scalars

- Vectors Vector quantity has magnitude and

direction - Vector represented by arrows with length equal to

vector magnitude and arrow direction giving the

vector direction - Example Displacement Vector
- Scalar Scalar quantity with magnitude only.
- Example Temperature,mass

Ch 3-3 Adding Vectors Geometrically

- Vector addition
- Resultant vector is vector sum of two vectors
- Head to tail rule vector sum of two vectors a

and b can be obtained by joining head of a

vector with the tail of b vector. The sum of

the two vectors is the vector s joining tail of a

to head of b - sa b b a

Ch 3-3 Adding Vectors Geometrically

- Commutative Law Order of addition of the vectors

does not matter - a b b a
- Associative Law More than two vectors can be

grouped in any order for addition - (ab)c a (bc)
- Vector subtraction Vector subtraction is

obtained by addition of a negative vector

Check Point 3-1

- The magnitude of displacement a and b are 3 m and

4 m respectively. Considering various orientation

of a and b, what is - i) maximum magnitude for c and ii) the minimum

possible magnitude?

- i) c-maxab347

a

b

c-max

ii) c-mina-b3-41

a

-b

c-min

Ch 3-4 Components of a Vector

- Components of a Vector Projection of a vector on

an axis - x-component of vector
- its projection on x-axis
- axa cos?
- y-component of a vector
- Its projection on y-axis
- aya sin?
- Building a vector from its components
- a ?(ax2ay2) tan ? ay/ax

Check Point 3-2

- In the figure, which of the indicated method for

combining the x and y components of the vector d

are propoer to determine that vector?

- Ans
- Components must be connected following

head-to-tail rule. - c, d and f configuration

Ch 3-5 Unit Vectors

- Unit vector a vector having a magnitude of 1 and

pointing in a specific direction - In right-handed coordinate system, unit vector i

along positive x-axis, j along positive y-axis

and k along positive z-axis. - a ax i ay j az k
- ax , ay and az are scalar components of the

vector - Adding vector by components r ab
- then rx ax bx ry ay by rz axz bz
- r rx i ry j rz k

Ch 3-6 Adding Vectors by components

- To add vectors a and b we must
- 1) Resolve the vectors into their scalar

components - 2) Combine theses scalar components , axis by

axis, to get the components of the sum vector r - 3) Combine the components of r to get the vector

r - r a b
- aaxi ay j b bxibyj
- rxax bx ry ay by
- r rx i ry j

Check Point 3-3

- Ans
- a) ,
- b) , -
- c) Draw d1d2 vector using head-to-tail rule
- Its components are ,

- a) In the figure here, what are the signs of the

x components of d1 and d2? - b) What are the signs of the y components of d1

and d2? - c) What are the signs of x and y components of

d1d2?

Ch 3-8 Multiplication of vectors

- Multiplying a vector by a scalar
- In multiplying a vector a by a scalar s, we get

the product vector sa with magnitude sa in the

direction of a ( positive s) or opposite to

direction of a ( negative s)

Ch 3-8 Multiplication of vectors

- Multiplying a vector by a vector
- i) Scalar Product (Dot Product)
- a.b a(b cos?)b(a cos?)
- (axiayj).(bxibyj)
- axbxayby
- where b cos? is projection of b on a and a

cos? is projection of a on b

Ch 3-8 Multiplication of vectors

- Since a.b ab cos?
- Then dot product of two similar unit vectors i or

j or k is given by - i.ij.jk.k1 (?0, cos?1)
- is a scalar
- Also dot product of two different unit vectors is

given by - i.jj.kk.i 0 (?90, cos?0).

Check Point 3-4

- Vectors C and D have magnitudes of 3 units and 4

units, respectively. What is the angle between

the direction of C and D if C.D equals - a) Zero
- b) 12 units
- c) -12 units?

- a) Since a.b ab cos? and a.b0 cos? 0 and ?

cos-1(0)90? - (b) a.b12, cos? 1 and
- ? cos-1(1)0?
- (vectors are parallel and in the same

direction) - (c) b) a.b-12, cos? -1 and
- ? cos-1(-1)180?
- (vectors are in opposite directions)

Ch 3-8 Multiplication of vectors

- Multiplying a vector by a vector
- ii) Vector Product (Cross Product)
- c ax b absin?
- c (axiayj)x(bxibyj)
- Direction of c is perpendicular to plane of a

and b and is given by right hand rule

Ch 3-8 Multiplication of vectors

- Since a x b ab sin ? is a vector
- Then cross product of two similar unit vectors i

or j or k is given by - ixi jxj kxk 0 (as ?0 so sin ?0).
- Also cross product of two different unit vectors

is given by - ixjk jxk i kxi j
- jxi -k kxj -i ixk-j

Ch 3-8 Multiplication of vectors

- If aaxi ayj and bbxibyj
- Then c axb
- (axi ayj )x(bxibyj)
- axi x (bxi byj) ayj (bxibyj)
- axbx(i x i ) axby(i x j) aybx(jx i )

ayby(j x j) - but ixi0, ixjk jxi-k
- Then caxb (axby-aybx) k

Check Point 3-5

- Vectors C and D have magnitudes of 3 units and 4

units, respectively. What is the angle between

the direction of C and D if magnitude of C x D

equals - a) Zero
- b) 12 units

a) Since a xb ab sin? and axb0 sin ? 0 and ?

sin-1 (0) 0?, 180? (b) a xb 12, sin ? 1 and

? sin-1(1)90?