VECTOR

1. 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. 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) • Vectorshave 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 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 = 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)

29. Cross Product aXb = IaIIbI SinӨ.n bXa = IbIIaI SinӨ.-n aXb = -bXa cross Product is not Commutative aXa =0 for every vector A as Ө = 0 ThereforeiXi =jXj = kXk = 0 iXj = k jXi = -k jXk = iandkXj = -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= ikXi = 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 perpto both a & b andc.d =15

39. VALUE BASED QUESTION :