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

COLLINEARITY OF VECTORS 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, 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.

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

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

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

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

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 :