### Chapter 3

```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.
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:
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
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
does not matter
a + b = b + a
 Associative Law: More
than two vectors can be
grouped in any order for
(a+b)+c= a + (b+c)
 Vector subtraction: Vector
subtraction is obtained by
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-max=a+b=3+4=7
a
b
c-max
ii) c-min=a-b=3-4=1
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
ax=a cos
 y-component of a vector:
Its projection on y-axis
ay=a sin
 Building a vector from its
components
a =(ax2+ay2); 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
 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= a+b
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
a=axi + ay j; ; b = bxi+byj
rx=ax + bx; ry = ay + by
r= rx i + ry j
Check Point 3-3
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 d1+d2?
Ans:
a) +, +
b) +, c) Draw d1+d2 vector
rule
Its components are
+, +
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)
= (axi+ayj).(bxi+byj)
= axbx+ayby
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.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).
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.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)
Ch 3-8 Multiplication of vectors
 Multiplying a vector by a
vector:
ii) Vector Product (Cross
Product)
c= ax b = absin
c = (axi+ayj)x(bxi+byj)
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:
ixj=k; jxk= i ; kxi =j
jxi= -k; kxj= -i ; ixk=-j
Ch 3-8 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
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 axb=0 sin
 =0 and = sin-1 (0) =0◦, 180◦
(b) a xb =12, sin  =1 and
= sin-1(1)=90◦
```