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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

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

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

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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.

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NOTATION OF VECTORS:

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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.

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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.

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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.

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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) (k 1 + k 2 ) a = k 1 a + k 2 a (Multiplication By Scalars) k(a + b) = k a + k b (Multiplication By Scalars)

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COMPONENT OF A VECTOR

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SOME OTHER PROPERTIES:

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COLLINEARITY OF TWO VECTORS:

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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

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VECTOR JOINING TWO POINTS:

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SECTION FORMULA : INTERNAL DIVISION

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EXTERNAL DIVISION

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

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

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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

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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.

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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.

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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 a x b= -b x a

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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].

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PROPERTIES

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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

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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 0 + a2b i.j = 0 = j.i a3b3 j.k = 0 = k.j a.b = a1b1 + a2b2 +a3b3 k.i = 0= i.k)

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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

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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

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

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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

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

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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

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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

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LEVEL 1 QUESTIONS:

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LEVEL 2 QUESTIONS: 1) 2) 3)

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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 Level 3

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VALUE BASED QUESTION :

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