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VECTOR
• TOPIC 1
• INTRODUCTION
• SOME BASIC CONCEPTS
• TYPES OF VECTORS
• ADDITION OF VECTORS
PRESENTED BY SATISH ARORA
• TOPIC 2
• DEFINITION AND PROPERTIES OF
•
DOT PRODUCTS
• CROSS PRODUCTS
PRESENTED BY SATISH ARORA
• TOPIC 3
• FORMULA FOR FINDING CROSS PRODUCT BETWEEN 2 VECTORS
• SCALER TRIPPLE PRODUCT
• PROBLEM RELATED TO ABOVE TOPICS
•
SUB TOPICS:
• INTRODUCTION
• SOME BASIC CONCEPTS
• TYPES OF VECTORS
• ADDITION OF VECTORS
• SECTION FORMULA
• COLLINEARITY OF VECTORS
PRE KNOWLEDGE:
• KNOWING ABOUT SCALAR QUANTITY
• KNOWING ABOUT MAGNITUDE AND DIRECTION
• DIRECTION RATIOS AND DIRECTION COSINE
• MASS,SPEED
WHAT ARE VECTORS AND SCALARS ??
• A vector has direction and magnitude both but scalar has only
magnitude.
• FOR EXAMPLE :
• Scalars only have only magnitude (ex. 50 m)
• Vectors have both magnitude and direction (ex. 50 m, North)
• Magnitude of a vector a is denoted by |a| or a. It is non-negative
scalar.
NOTATION OF VECTORS:
TYPES OF VECTORS :
• Zero or Null Vector: A vector whose initial and terminal points are
coincident is called zero or null vector. It is denoted by 0.
• Unit Vector: A vector whose magnitude is unity is called a unit vector
in direction of a which is denoted by a i.e a =a/|a|
• Collinear or Parallel Vectors: Two or more vectors are collinear if they
are parallel to the same line , irrespective of their magnitude and
directions.
• Coinitial Vectors: Vectors having same initial point are called coinitial
vectors.
• Equality of Vectors: Two 𝑣𝑒𝑐𝑡𝑜𝑟𝑠 a and b are said to be equal written
as a = b, if they have (i) same length (ii) the same or parallel support
and (iii) the same sense.
• Negative of a Vector: A vector having the same magnitude as that of
a given vector a and the direction opposite to that of a is called the
negative of a and it is denoted by —a.
ADDITION OF TWO VECTORS:
• TRIANGLE LAW OF VECTOR ADDITION
• Let a and b be any two vectors. From the terminal point of a, vector b
is drawn. Then, the vector from the initial point O of a to the terminal
point B of b is called the sum of vectors a and b and is denoted by a +
b. This is called the triangle law of addition of vectors.
• PARALLELOGRAM LAW OF VECTOR ADDITION:
• Let a and b be any two vectors. From the initial point of a, vector b is
drawn and parallelogram OACB is completed with OA and OB as
adjacent sides. The vector OC is defined as the sum of a and b. This is
called the parallelogram law of addition of vectors.
• The sum of two vectors is also called their resultant and the process
of addition as composition.
Properties of Vector Addition
• a + b = b + a (Commutativity)
• a + (b + c)= (a + b)+ c (Associativity)
• a+ O = a (Additive Identity)
• a + (— a) = 0 (Additive Inverse)
• (k1 + k2) a = k1 a + k2a (Multiplication By Scalars)
• k(a + b) = k a + k b (Multiplication By Scalars)
COMPONENT OF A VECTOR
SOME OTHER PROPERTIES:
COLLINEARITY OF TWO VECTORS:
POSITION VECTOR OF A POINT:
• The position vector of a point P with respect to a fixed point, say O, is
the vector OP. The fixed point is called the origin. Let PQ be any
vector. We have
PQ = PO + OQ = — OP + OQ = OQ — OP = Position vector of Q —
Position vector of P.
i.e., PQ = PV of Q — PV of P
VECTOR JOINING TWO POINTS:
SECTION FORMULA :
• INTERNAL DIVISION
• EXTERNAL DIVISION
SUBTOPICS
• TYPES OF PRODUCTS (DEFINITION AND PROPERTIES)
• Dot Products
• Cross Products
PRE - REQUISITES
•
•
•
•
•
Physical Quantity
Type of physical Quantities
Position Vector
Unit Vectors
Representation of Vectors
SCALAR PRODUCT
The scalar product of two nonzero vectors a and b is denoted by a⋅
b, defined as
a ⋅ b = | a | | b | cosθ,
where, θ is the angle between a and b,
If either a = 0 or b = 0,
then θ is not defined and 0 ≤ θ ≤ π
PROPERTIES
• a.b=0 if a and b are perpendicular
• If θ= 0 then a ⋅ b = | a | | b | .
a ⋅a =| a |^2 , as θ in this case is 0
• If θ = π, then a ⋅b = −| a | | b |
• In view of the Observations 2 and 3, for mutually perpendicular unit
vectors
• iˆ, ˆj and kˆ, we have iˆ ⋅ iˆ = ˆj ⋅ ˆj = kˆ ⋅ kˆ =1,
iˆ ⋅ ˆj = ˆj ⋅ kˆ = kˆ iˆ= 0.
• Cos θ= a.b/ [a] [b]
 Commutative law a.b=b.a.
VECTOR PRODUCT OR CROSS PRODUCT
The vector product of two nonzero vectors a and b , is denoted by aX b and defined as,
×b = | a || b | sin θ nˆ
a
where, θ is the angle between a and b , 0 ≤ θ ≤ π and ˆ n is
a unit vector perpendicular to both a and b , such that a, b and nˆ form a right handed system .
i.e., the right handed system rotated from a to b moves in the direction of ˆ n . If either a = 0
or b = 0 ,
then θ is not defined and in this case, we define a × b = 0.
PROPERTIES
• a x b is a vector.
• If a and b be two nonzero vectors. Then a × b = 0 if and only if a and b
are parallel (or collinear) to each other, i.e., θ=0.
• If θ= 90 then, axb= | a | | b |.
• for mutually perpendicular unit vectors iˆ, ˆj and kˆ , we have
ixi=jxj=kxk=0
iˆ× ˆj = kˆ, ˆj × kˆ = iˆ, kˆ ×iˆ = ˆj
• axb= -bxa
PROPERTIES
• If vector a and b are representing the adjacent side of a triangle then,
area of triangle = ½ [axb].
• If vector a and b are representing the adjacent side of a
parallelogram then, area of //gram= [axb].
PROPERTIES
i
a  b  a1
b1
j
a2
b2
k
a3
b3
Resolution of Vector in 2-D Plane & 3D Space
OP = OQ + QP
R = xi + yj
i, j are unit vector along
coordinate axes
Similarly any vector in space can be expressed in terms of i, j, k
R = xi + yj + zk
i.e.
Formula for Finding Dot Product of 2
Vector in 3-D Space
a = a1j + a2j + a3k
b = b1i + b2j + b3k
a.b = (a1i + a2j + a3k).(b1i + b2j + b3k)
= a1b1i.i + a1b2i.j + a1b3i.k +
a2b1j.i + a2b2j.j + a2b3j.k +
a3b1k.i + a3b2k.j + a3b3k.k
(As
= a1b1 + 0
+0
+
i.i = j.j = k.k =1
0
+ a2b2 + 0
+
i.j = 0 = j.i
0
+0
+ a3b3
j.k = 0 = k.j
a.b = a1b1 + a2b2 +a3b3
k.i = 0= i.k)
Cross Product
aXb = IaI IbI SinӨ.n
bXa = IbI IaI SinӨ.-n
aXb = -bXa
cross Product is not Commutative
aXa =0 for every vector
A as Ө = 0
Therefore
iXi =jXj = kXk = 0
iXj = k
jXi = -k
jXk = i
and kXj = -i
kXi = j
iXk = -j
Formula for finding Cross Product
between 2 vectors
a = a1i + a2j + a3k
b = b1i +b2j +b3k
aXb = (a1i +a2j +a3k)X(b1i + b2j + b3k)
= a2b1iXi + a1b2iXj + a1b3iXk +
= a2b1jXi + a2b2jXj + a2b3jXk +
= a3b1kXi + a3b2kXj + a3b3kXk
Now iXi = jXj = kXk =0
iXj = k
jXi = -k
jXk = i
kXj = -I
kXi = j
iXk = -j
Continued…
So
aXb = 0 + a1b2k - a1b3j a2b1k + 0 + a2b3i +
a3b1j - a3b2i + 0
=> aXb = i(a2b3 - a3b2) - j(a1b3 - a3b1)+
k(a1b2 - a2b1)
i
a  b  a1
b1
j
a2
b2
k
a3
b3
Scalar Tripple Product (Mixed
Product)
For three vectors a, b and c
a.(bXc) is defined and
aX(b.c) is not defined
Also a.(bXc) = (aXb).c = [a b c]
Is the mixed product of 3 vectors
Geometrical Significance of Mixed
Product
Geometrical Significance of mixed product [a b c] represents volume of
parallelopiped determined by vectors a, b and c
Continued…
Vectors a, b, c will be coplaner if [a b c]
If a = a1i + a2j + a3k
b = b1i + b2j + b3k
c = c1i + c2j + c3k
Then
[a b c] = det a1 a2 a3
b1 b2 b3
c1 c2 c3
Suggested Problems
Level 1
Ques 1. Find projection of
a = 3i – 2j + 3k
on b = I + j + k
Ques 2. Find p if
a = 2i – 4j + k is perp to
b = i+ pj + 2k
Ques 3. Find unit vector along the
direction of
-3i + 2j + 5k
LEVEL 1 QUESTIONS:
LEVEL 2 QUESTIONS:
1)
2)
3)
Level 3
Ques 1. i, j, k are unit vector along
coordinate axes
a = 3i – j
b = 2i + j – 3k
Then express b = b1 + b2 such that
b1IIa and b2 perp b
Ques 2. If a, b, c are mutually perpendicular
vectors of equal magnitude show
that a+ b + c is equally inclined to
a, b and c
Ques 3. Let a = i + 4j + 2k
b = 3i – 2j + 7k
c = 2i – j + 4k
For vector d perp to both a & b and c.d =15
VALUE BASED QUESTION :