1. PRESENTED BY SATISH ARORA
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

2. COLLINEARITY OF VECTORS
SUB TOPICS:
INTRODUCTION
SOME BASIC CONCEPTS
TYPES OF VECTORS
ADDITION OF VECTORS
SECTION FORMULA
COLLINEARITY OF VECTORS

3. PRE KNOWLEDGE: KNOWING ABOUT SCALAR QUANTITY
KNOWING ABOUT MAGNITUDE AND DIRECTION
DIRECTION RATIOS AND DIRECTION COSINE
MASS,SPEED

4. 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.

5. NOTATION OF VECTORS:

6. 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.

7. 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.

8. 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.

9. 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.

10. 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)

11. COMPONENT OF A VECTOR

13. SOME OTHER PROPERTIES:

14. COLLINEARITY OF TWO VECTORS:

15. 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

16. VECTOR JOINING TWO POINTS:

17. SECTION FORMULA :
INTERNAL DIVISION

18. EXTERNAL DIVISION

19. SUBTOPICS TYPES OF PRODUCTS (DEFINITION AND PROPERTIES) Dot Products
Cross Products

20. PRE - REQUISITES Physical Quantity Type of physical Quantities
Position Vector
Unit Vectors
Representation of Vectors

21. 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

22. 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.

23. VECTOR PRODUCT OR CROSS PRODUCT
The vector product of two nonzero vectors a and b , is denoted by aX b and defined as, a ×b = | a || b | sin θ nˆ 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.

24. 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

25. 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].

26. PROPERTIES

27. Resolution of Vector in 2-D Plane & 3-D 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 i.e.
R = xi + yj + zk

28. 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
= a1b i.i = j.j = k.k =1
a2b i.j = 0 = j.i
a3b j.k = 0 = k.j
a.b = a1b1 + a2b2 +a3b k.i = 0= i.k)

29. 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

30. 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 jXk = i kXi = j
jXi = -k kXj = -I iXk = -j

31. Continued… So aXb = 0 + a1b2k - a1b3j - a2b1k + 0 + a2b3i +
a3b1j - a3b2i + 0
=> aXb = i(a2b3 - a3b2) - j(a1b3 - a3b1)+
k(a1b2 - a2b1)

32. 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

33. Geometrical Significance of Mixed Product
Geometrical Significance of mixed product [a b c] represents volume of parallelopiped determined by vectors a, b and c

34. 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

35. 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

36. LEVEL 1 QUESTIONS:

37. LEVEL 2 QUESTIONS: 1) 2) 3)

38. 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

39. VALUE BASED QUESTION :