Lecture 2

Vectors

AgendaAgenda

Chapter 2Chapter 2

Vector sVector sAdditionAddition subtraction subtraction componentscomponents scalar productscalar product cross productcross product

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

vectors

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

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

s=a + b = b + a

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(a+b)+c= a + (b+c)

Vector subtraction: Vector subtraction is obtained by addition of a negative vector

Adding Vectors Geometrically

Exemple

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-max=a+b=3+4=7

a b

c-max

ii) c-min=a-b=3-4=1

a -b

c-min

Components of a Vector

Components of a Vector: Projection of a vector on an axis

x-component of vector: its projection on x-axis

ax=a cos y-component of a vector: Its projection on y-axis ay=a sin Building a vector from its

components a =(ax

2+ay2); tan =ay/ax

Exemple

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

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= a+b

then rx= ax + bx ; ry= ay + by ; rz= axz+ bz

r = rx i+ ry j + rz k

Adding Vectors by components

To add vectors a and b we must:1) Resolve the vectors into their scalar components2) Combine theses scalar components , axis by

axis, to get the components of the sum vector r3) Combine the components of r to get the vector r

r= a + ba=axi + ay j; ; b = bxi+byj

rx=ax + bx; ry = ay + by

r= rx i + ry j

Exemplea) 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 d1+d2?

Ans:a) +, +b) +, -c) Draw d1+d2

vector using head-to-tail rule

Its components are +, +

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)

Multiplication of vectors

Multiplying a vector by a vector:

i) Scalar Product (Dot Product)

a.b= a(b cos)=b(a cos) = (axi+ayj).(bxi+byj)

= axbx+ayby

where b cos is projection of b on a and a cos is projection of a on b

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.i=j.j=k.k=1 (=0, cos=1) is a scalar Also dot product of two different

unit vectors is given by: i.j=j.k=k.i =0 (=90, cos=0).

Exemple

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.b=0 cos =0 and = cos-

1(0)=90◦; (b) a.b=12, 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)

Multiplication of vectors

Multiplying a vector by a vector:

ii) Vector Product (Cross Product)

c= ax b = ab sin c = (axi+ayj) x (bxi+byj)

Direction of c is perpendicular to plane of a and b and is given by right hand rule

Multiplication of vectors

Since a x b= ab sin is a vectorThen 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:

ixj=k; jxk= i ; kxi =j jxi= -k; kxj= -i ; ixk=-j

Multiplication of vectors

If a=axi +ayj and b=bxi+byj Then c = axb =(axi +ayj ) x (bxi+byj)

= axi x (bxi +byj) + ayj (bxi+byj)

= axbx(i x i ) + axby(i x j) + aybx(jx i ) + ayby(j x j)

but ixi=0, ixj=k; jxi=-k Then c=axb = (axby-aybx) k

Exemple

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) Zerob) 12 units

a) Since a xb= ab sin and axb=0 sin =0 and = sin-1 (0)

=0◦, 180◦

(b) a xb =12, sin =1 and

= sin-1(1)=90◦