Title XII - maths - chapter 9 - ADDITION OF VECTORS (11.03.2015)(1-34).pdf ✅

1 JEE MAINS - VOL - I ADDITION OF VECTORS  Definition of Vector and Scalar:  Scalar: A physical quantity which has only magnitude is called a scalar. Examples:Length,volume, speed, time.  Vector: A vector is a physical quantity which has both magnitude and direction . Geometrically a directed line segment is called a vector. Examples: Force, Velocity, acceleration. Note: All real numbers are scalars.  Notation: Vectors are denoted by directed line segments such as ABCD.... or by a b, ... If AB is a vector then A is called initial point and B is called the terminal point or final point and the direction of AB is from A to B. The magnitude of AB is denoted by AB or AB and, is the distance between the points A and B.  Types of Vectors:  Position Vector: Let O be a fixed point (called the origin ) and let P be any point. If OP r  then r is called the position vector of P with respect to O.  Null Vector: A vector having zero magnitude (arbitrary direction) is called the null (zero) vector. It is denoted by 0 . Note (i ) A zero vector can be regarded as having any direction for all mathematical calculations. (ii) A non-zero vector is called a proper vector  Unit Vector: A vector whose magnitude is equal to one unit is called a unit vector.  Localised vector :A vector is a localised vector, if the vector is specified by giving either initial point or terminal point (or) if a vector is specified by fixing atleast one of its ends is called a localised vector.  Free Vector: When a vector is specified by not fixing initial point or terminal point or both, is called a free vector, i.e., a free vector does not have specific initial point or terminal point or both. Note: (i) A vector a means we are free to choose initial or terminal point anywhere. Once initial point is fixed at A then terminal point is uniquely fixed at B such that AB a  (ii) A free vector is subjected to parallel displacement without changing the magnitude and direction. (iii) In general vectors are considered to be free vectors unless they are localised.  Let a be a nonzero vector then (i) Unit vector in the direction of a a a  (ii) Unit vector in the direction opposite to that of a is a a  (iii) Unit vectors parallel to a a a   . (iv) The vectors having magnitude m units and parallel to ma a a   . WE-1: a i j k    2 2 and b i j k    3 6 2 , then vector in the direction of a and having magnitude as b is Sol: The required vector is   7 2 2 3 b a i j k a     Equal Vectors: Two vectors a and b are equal if they have the same magnitude i e a b .   and they are in the same ADDITION OF VECTORS SYNOPSIS
ADDITION OF VECTORS 2 JEE MAINS - VOL - I direction.  Let a and b be the position vectors of the points A and B respectively. Then AB  (position vector of B) - (position vector of A) i.e., AB b a    Co-Initial Vectors: Vectors having the same initial point are called coinitial vectors. The vectors OA OB OC , , ... are coinitial vectors.  Co-Terminal Vectors: The vectors having the same terminal point are called the co-terminal vectors. The vectors AO BO CO , , ... are co-terminal vectors.  Addition of Vectors:  Triangle law of Vector Addition: A B C a b a  b If AB a  and BC b  are two non-zero vectors are represented by two sides of a triangle ABC then the resultant (sum) vector is given by the closing side  AC of the triangle in opposite direction. i.e., AC AB BC a b      Parallelogram Law of Vector Addition: O B C a A b a  b If a and b are two adjacent sides of the parallelogram, then their sum (resultant) a b  represents the diagonal of the parallelogram through the common points. It is known as parallelogram law of vector addition. i.e., OC OA OB a b      Properties of Addition of Vectors: i) Addition of vectors is commutative i.e., a b b a    ii) Addition of vectors is associative i.e., a b c a b c         iii) There exists a vector 0 such that a a a     0 0 . Then 0 is called the additive identity vector. iv) To each vector a there exists a vector a such that a a a a           0 .Then a is called the additive inverse of a .  Scalar Multiplication of Vectors: Let a be a nonzero vector and let m be a scalar. Then ma is a scalar multiplication of a by m . Note: i) The direction of ma is along a if m>o. ii) The direction of ma is opposite to that of a if m3 JEE MAINS - VOL - I ADDITION OF VECTORS  Components of a space Vector: O Z P( , , ) x y z X Y i k j  Let i j k , , be unit vectors acting along the positive directions of x, y and z axes respectively, then position vector of any point P in the space is OP xi yj zk    . Here (x, y, z) are called scalar components of vector OP along respective axes and xi yj zk , , are called vector components of OP along respective axes & 2 2 2 OP x y z     Section Formula : If the position vectors of the points A, B w.r.t. O are a and b and if the point C divides the line segment AB in the ratio m n: internally m n   0, 0, then the position vector of C is mb na OC m n    .  If C is an external point that divides A a B b  ,   in the ratio m n: externally then mb na OC m n    m n   and m n, 0    The point C (mid point) divides A a B b  ,   in the ratio 1:1 , then 2 a b OC   .  Points of trisection : Two points which divide a line segment in the ratio 1: 2 and 2 :1 are called the points of trisection. WE-2: If a b, are the position vectors of A,B respectively and C is a point on AB produced such that AC = 3AB, then the position vector of C is Sol: Let the position vector for C be c , then B divides AC internally in the ratio 1:2, therefore 2 3 2 . 2 1 a c b c b a        The position vector of the centroid G of the triangle ABC with vertices a b c , , is 3 a b c   and the incentre aa bb cc I a b c      , where a BC b CA   , and c AB  .  In ABC if D, E, F are the mid points of the sides BC, CA, AB respectively and G is the centroid then (i)GA GB GC    0 (ii) AD BE CF    0. (iii) OA OB OC OD OE OF OG       3  If a b c , , and d are the position vectors of the vertices A, B, C and D respectively of a tetrahedron ABCD then the position vector of its centroid is 4 a b c d     Angle between two vectors : If OA a OB b   , be two non-zero vectors and 0 0     AOB   , 0 180 is defined as the angle between a and b and is written as a b,  . A A B O   180 -O  180 -O  B a a b b
ADDITION OF VECTORS 4 JEE MAINS - VOL - I Note: If a  0 or b  0 , then angle between a and b is undefined. Note: (i) a b, 0    a and b are like vectors. (ii) ,  2 a b    a and b are orthogonal vectors. (iii)a b,    a and b are unlike vectors. (iv)a b b a , ,     and a b a b , ,       (v)ma nb a b , ,    (if m,n have same signs). (vi)    0 ma nb a b , 180 ,   (if m,n haveopposite signs).  Right handed and left handed triads Let OA a OB b OC c    , , be three non- coplanar vectors. Viewing from the point C, if the rotation of OA to OB does not exceed angle 180 in anti-clock sense, then a b c , , are set to form a right handed system of vectors and we say simply that ( , , ) a b c is right handed vector triad. If ( , , ) a b c is not a right handed system, then it is called left handed system. A A O O B B C C a a c c b b  Direction cosines and Direction ratios of a vector: Let i j k , , be an unit vector traid in the right handed system and r is a vector. If     r i r j , , ,    and   r k, , then cos ,cos ,cos    are called the direction cosines of r denoted by l m n , , respectively.The numbers proportional to direction cosines of a given vector, i,e., kl km kn , , are called the direction ratios of that vector for k R  0  Some important results: If l m n , , are the direction cosines of a line, then 2 2 2 l m n   1.  If OP r  and P is the ordered triad  x y z , ,  then x r lr y r mr     cos , cos   and z r nr   cos .  The direction cosines of the vectors i j k , , are respectively 1,0,0 , 0,1,0 , 0,0,1      .  Linear Combination of Vectors:  Linear Combination: Let 1 2 , ,..., n a a a be n vectors and let r be any vector. Then 1 1 2 2 3 3 .... n n r x a x a x a x a      is called a linear combination of the vectors 1 2 , ,..., n a a a .  Collinear Vectors: Vectors which lie on a line or on parallel lines are called collinearvectors (whatever be their magnitudes). Note:(i) Two vectors a and b are collinear if and only if a mb  or b na  where m ,n are scalars (real numbers). (ii) Let the vectors a a i a j a k    1 2 3 , b b i b j b k    1 2 3 are collinear if and only if 3 3 2 2 1 1 b a b a b a   (iii) The Vectors a b, are collinear vectors iff   0 a b, 0 or180    Let a and b be two non collinear vectors and let r be any vector coplanar with them.Then r xa yb   and the scalars x and y are unique in the sense that if 1 1 r x a y b   and 2 2 r x a y b   then 1 2 1 2 x x y y   and . The vector equation r xa yb   implies that