Presentation on theme: "PRESENTED BY SATISH ARORA"— Presentation transcript:
1PRESENTED BY SATISH ARORA VECTORTOPIC 1INTRODUCTIONSOME BASIC CONCEPTSTYPES OF VECTORSADDITION OF VECTORS PRESENTED BY SATISH ARORATOPIC 2DEFINITION AND PROPERTIES OFDOT PRODUCTSCROSS PRODUCTSPRESENTED BY SATISH ARORATOPIC 3FORMULA FOR FINDING CROSS PRODUCT BETWEEN 2 VECTORSSCALER TRIPPLE PRODUCTPROBLEM RELATED TO ABOVE TOPICS
2COLLINEARITY OF VECTORS SUB TOPICS:INTRODUCTIONSOME BASIC CONCEPTSTYPES OF VECTORSADDITION OF VECTORSSECTION FORMULACOLLINEARITY OF VECTORS
3PRE KNOWLEDGE: KNOWING ABOUT SCALAR QUANTITY KNOWING ABOUT MAGNITUDE AND DIRECTIONDIRECTION RATIOS AND DIRECTION COSINEMASS,SPEED
4WHAT 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.
6TYPES 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.
7Equality 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.
8ADDITION OF TWO VECTORS: TRIANGLE LAW OF VECTOR ADDITIONLet 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.
9PARALLELOGRAM 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.
10Properties 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)
15POSITION 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 havePQ = PO + OQ = — OP + OQ = OQ — OP = Position vector of Q — Position vector of P.i.e., PQ = PV of Q — PV of P
19SUBTOPICS TYPES OF PRODUCTS (DEFINITION AND PROPERTIES) Dot Products Cross Products
20PRE - REQUISITES Physical Quantity Type of physical Quantities Position VectorUnit VectorsRepresentation of Vectors
21SCALAR PRODUCTThe 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
22PROPERTIES 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 0If θ = π, then a ⋅b = −| a | | b |In view of the Observations 2 and 3, for mutually perpendicular unit vectorsiˆ, 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.
23VECTOR 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.
24PROPERTIES 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 haveixi=jxj=kxk=0iˆ× ˆj = kˆ, ˆj × kˆ = iˆ, kˆ ×iˆ = ˆjaxb= -bxa
25PROPERTIESIf 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].
29Cross Product aXb = IaI IbI SinӨ.n bXa = IbI IaI SinӨ.-n aXb = -bXa cross Product is not CommutativeaXa =0 for every vectorA as Ө = 0Therefore iXi =jXj = kXk = 0iXj = k jXi = -kjXk = i and kXj = -ikXi = j iXk = -j
32Scalar Tripple Product (Mixed Product) For three vectors a, b and ca.(bXc) is defined andaX(b.c) is not definedAlso a.(bXc) = (aXb).c = [a b c]Is the mixed product of 3 vectors
33Geometrical Significance of Mixed Product Geometrical Significance of mixed product [a b c] represents volume of parallelopiped determined by vectors a, b and c
34Continued… Vectors a, b, c will be coplaner if [a b c] If a = a1i + a2j + a3kb = b1i + b2j + b3kc = c1i + c2j + c3kThen [a b c] = det a1 a2 a3b1 b2 b3c1 c2 c3
35Suggested Problems Level 1 Ques 1. Find projection of a = 3i – 2j + 3k on b = I + j + kQues 2. Find p ifa = 2i – 4j + k is perp tob = i+ pj + 2kQues 3. Find unit vector along thedirection of-3i + 2j + 5k
38Level 3Ques 1. i, j, k are unit vector alongcoordinate axesa = 3i – jb = 2i + j – 3kThen express b = b1 + b2 such thatb1IIa and b2 perp bQues 2. If a, b, c are mutually perpendicularvectors of equal magnitude showthat a+ b + c is equally inclined toa, b and cQues 3. Let a = i + 4j + 2kb = 3i – 2j + 7kc = 2i – j + 4kFor vector d perp to both a & b and c.d =15