INTRODUTION TO VETOR LULUS LSSIFITION OF VETORS roadly vectors can be classified into two categories (i) (ii) xial Vectors : Where a vector has rotational motion lying along the normal to the plane of rotation of the body and remains unchanged under inversion. e.g.: Torque, angular momentum etc. Polar Vector : Where a vector has linear motion in a particular direction but changes under inversion or reflection. e.g. displacement, position vector, velocity etc. Some special vectors (i) Unit vector : It is a vector with unit magnitude and characterizes the direction of the vector mathematically it is denoted by  = In artesian coordinate system, let us choose three unit vectors along three mutually perpendicular axes as ˆi, ˆj and ˆk in x, y and z directions respectively. Then any arbitrary vector can be expressed as = x i ˆ ˆ ˆ y j zk where x, y and z are called the components of in x, y and z directions.
(2) Introduction to Vector alculus y y ˆk ĵ î x ˆ zk 0 ˆ y j x z z xˆ i The magnitude of is given using parallelogram law = hence unit vector along x y z is given by xiˆ ˆ y j ˆ zk  = Direction cosines The cosines of the angles, which makes with x, y and z axis are called direction cosines of the vector. If l, m and n are the direction cosines along ox, oy and oz axes, then l = x x cos or l m = n = y cos or y m z cos or z n and l m n = 2 x y z 2 2 1
Introduction to Vector alculus (3) so l m n = cos cos cos 1 li ˆ m ˆ j n k ˆ and = and the unit vector â = liˆ mˆj n kˆ (ii) (iii) Null Vector : ny vector with magnitude zero is called null vector. It is collinear with every vector and denoted by O. ollinear or parallel vector : When vectors are parallel, then these are collinear vectors, whatsoever their magnitudes may be. Direction of there vectors may be some or opposite. (iv) (v) (vi) D When any scalar is multiplied to any vector then the resultant vector becomes collinear with original one. e.g. i.e. vector is times with same direction as of. oplanar vectors : When vectors lies in the same geometrical plane they are called coplanar vectors. Otherwise these are called non-coplanar vectors. Like vectors : The collinear vectors with same sense of direction irrespective of magnitude are called like vectors. Reciprocal vectors : When the magnitude of a vector is reciprocal to the magnitude of other vector with same direction then it is called reciprocal vector. It is written as 1 i.e. 1 â where â is the unit vector along the direction of. Product of vectors (i) Scalar product or dot product : When the result of product of two vectors is a scalar quantity then this product is known as scalar (or dot) product of the given vectors.
(4) Introduction to Vector alculus Mathematically it is obtained by multiplying the magnitudes of the vectors with cosines of the angle between them i.e.. = cos. = cos P 0 y cos Q lternatively scalar product may be defined as multiplication of one vector with component of another in the direction of first. In case of artesian unit vectors ˆi.i ˆ = i. i cos0 1.1 1 ˆj. ˆj k ˆ. kˆ and ˆi. ˆj = i j cos 90 0 ˆj. kˆ i ˆ. kˆ if = x i ˆ ˆ ˆ y j z k and = ˆ x i ˆ ˆ y j z k then. x x y y z z (a) Scalar product obeys commutative law i.e... (b) It obeys distributive law i.e.... (c) Two non-zero vectors are orthogonal or perpendicular when = 90 i.e. cos = 0 then. = 0 similarly two vectors are collinear when = 0 or i.e. cos = ± 1 then for = 0,. = and for =,. = Physical examples (i) Work done W F. ds
Introduction to Vector alculus (5) (ii) (ii) Power = F.v (iii) Magnetic flux of a magnetic field =. ds where is magnetic flux density over an area ds. (iv) Electric flux of an electric field = E. ds where E the electric field intensity through elementary area ds. Vector product or cross product When the product of two vectors is a vector quantity, then the product is called vector product or cross product mathematically it is written as = = sin. nˆ where 0 here ˆn is the unit vector in the direction of normal to the plane containing and such that, and from a right handed coordinate system with rotation from to. for ˆi, ˆj and k ˆ. ˆi ˆi = i i sin 0 0 ˆj ˆj kˆ kˆ and ˆi ˆj = i j sin90 1 ˆj kˆ kˆ ˆi but ˆj ˆi = j i sin 90 1 kˆ ˆj ˆi kˆ if = ˆ xi ˆ ˆ y j z k and = xi ˆ ˆ ˆ y j z k then = sin n ˆ or ˆ ˆ ˆ i j k = = yz zy iˆ ˆ xz zx j xy yx
(6) Introduction to Vector alculus and ˆn = hence sin = where = and = x y z x y z if the rotation from to is anti clockwise then is +ve. and if rotation is clockwise then is ve. 0 Properties (a) ross product is not commutative but (b) It is distributive i.e. (c) If two vectors are collinear or parallel then = 0 or then sin = 0 and = 0 (d) Two vectors are perpendicular then = 90 so = nˆ
Introduction to Vector alculus (7) Some examples (i) Moment of forces = r F L r p m r v (iii) Linear velocity v r (ii) ngular momentum (iv) (v) Scalar Triple product force on a charged particle F = qv where q is in coulombs. Force on a charged particle moving through electric and magnetic field is q E v. this is known as Lorentz force. F = When a vector is scalarly multiplied with the cross product of other two vectors then the result is called scalar triple product. =.... = Vector triple product When any vector is vectorily multiplied with vector product of other two vectors taken in cyclic order then the result is known as vector triple product... Properties = =.. =.. 0 (i) (ii)..
(8) Introduction to Vector alculus (iii). D..D.D. D. D. D (iv) Vector differentiation This is the limiting value of ratio of a vector to the change of a scalar as the change tends to zero is called vector differentiation. f (u + u) f = f (u + u) f (u) Properties (a) f (u) df f = lim du u 0 u u u f u = lim u 0 u d d d du du du d d... d du du du d d d du du du d d d d.... du du du du d d d d du du du du d da ds as dt ds dt (b) (c) (d) (e) (f)
Introduction to Vector alculus (9) d de e.e e. de e 2e de dt dt dt dt d 2 de e 2e. dt dt d da at dt dt (g) (h) or