Tips and Formulas Vectors

VECTOR 1. Vector Quantities A physical quantity which requires magnitude and a particular direction, when it is expressed. 2. r r r R  A B Triangle law of Vector addition R  A2  B 2  2 AB cos90o  tan   B sin  A  B cos  If A = B then R  2 A cos  2 Rmax = A + B for  = 0° ; 3. &  2 Rmin = A – B for  = 180° Parallelogram law of Addition of Two Vectors If two vectors are represented by two adjacent sides of a parallelogram which are directed away from their common point then their sum (i.e. resultant vector) is given by the diagonal of the parallelogram passing away through that common point. r r r uuur uuur uuur r 2 2 AB  AD  AC  R or A+B = R  R = A B  2ABcos tan   4. B sin  and A  B cos  tan   Vector subtraction r r r R  AB r r r  R  A  B   R  A2  B 2  2 AB cos 90o, If A = B then R  2 A sin 5. A sin  B  A cos  tan   B sin  A  B cos   2 Addition of More than Two Vectors (Law of Polygon) If some vectors are represented by sides of a polygon in same order, then their resultant vector is represented by the closing side of polygon in the opposite order. D C C B A R B D A r r r r r R=A+B+C+D BASIC MATHS & VECTOR 6. Rectangular component of a 3-D vector (a) r A =A x ˆi + A y ˆj + A z kˆ Angle made with x - axis cos   Angle made with y - axis cos   Angle made with z - axis cos   (b) Az  A Ay A Az A  Ay2  Az2 Ay l 2 x  Ax2  Ay2  Az2 Az  A Az 2 x A  Ay2  Az2 m n , m, n are called direction cosines. l 2  m2  n 2  cos 2   cos 2   cos 2    Ax2  Ay2  Az2 Ax2  Ay2  Az2  2 1 or sin 2   sin 2   sin 2   2 7. General Vector in x-y plane r r  xiˆ  yjˆ  r cos  iˆ  sin  ˆj  8.  Scalar product (Dot Product) (a) (b) r r r r 1  A.B   cos    A  B  AB cos  Þ Angle between two vectors  AB  r r If A  A xˆi  A yˆj  A z kˆ and B  B xˆi  B yˆj  B z kˆ then r r r r A  B  A x B x  A y B y  A z B z and angle between A & B is given by cos   r r A.B  AB A x B x  A y B y  A z Bz A 2x  A 2y  A 2z B2x  B2y  B2z b (c) ˆi  ˆi  ˆj  ˆj  kˆ  kˆ  1 ˆi  ˆj  ˆj  kˆ  kˆ  ˆi  0 bx (d) rr r rr r Component of vector b along vector a,bij  b.a a (e) r r r rr r r r r Component of b perpendicular to a , b   b  b ij  b  b.a a     a 9. Cross Product (Vector product) r r r r A  B  AB sin  nˆ where n̂ is a vector perpendicular to A & B or their plane and its direction given by right hand thumb rule. A× B (a) A A B B (b) ˆ A z  ˆi(A y B z  A z B y )  ˆj(A x B z  A z B x )  k(A xBy  AyBx ) Bx By Bz (d) (e) iˆ  iˆ  0, ˆj  ˆj  0, kˆ  kˆ  0 (f) iˆ  ˆj  kˆ ; ˆj  kˆ  iˆ positive ^ i negative ^ k kˆ iˆ  ˆj ; ˆj iˆ  kˆ kˆ ˆj iˆ ; iˆkˆ ˆj r r d r r dA r dB A.B   A. dt dt dt   (ii) r r d r r dA r r dB A B   B  A. dt dt dt   When a particle moved from (x1, y1, z1) to (x2, y2, z2) then its displacement vector. r r r r = r2  r1  x2 iˆ  y2 ˆj  z2 kˆ  x1iˆ  y1 ˆj  z1kˆ    Lami’s theorem A A c B B b C a C sin A sin B Sin C = = a b c (x2,y2,z2) r1 r r | r | ( x2  x1 ) 2  ( y2  y1 ) 2  ( z2  z1 ) 2 Magnitude F F1 F = 2 = 3 sin 1 sin 2 sin r (x1,y1,z1)   ( x2  x1 )iˆ  ( y2  y1 ) ˆj  ( z2  z1 )kˆ 12. ^ j Differentiation (i) 11. Ay r r r r A  B  B  A r r r r r r (A×B).A  (A×B).B=0 (c) 10. ˆj kˆ ˆi r r A  B  Ax r2 BASIC MATHS & VECTOR r r A B (a) Area of triangle , Area = (c) For Parallel vectors, 1 AB sin  2 2 r r r A B  0 (e) For coplanar vectors , r r r A.B  0 13. Examples of dot products : (c) r r Work, W  F .d  Fd cos  r r Power, P  F .v  Fv cos  r r Electric flux, E = E.A  EA cos  (d) r r Magnetic flux, B = B.A  BA cos  (a) (b) (e) 14. r r (b)Area of parallelogram, Area = A  B  AB sin   Potential energy of dipole in r r uniform field, U   p.E r r r (d)For perpendicular vectors, A.B  0 r r r (f)For coplanar vectors, A. B  C  0  where F  force, d  displacement where F  force, v  velocity where E  electric flux, A  Area where B  magnetic field, A  Area where p  dipole moment, where E  Electric field  Examples of cross products : (a) r r r Torque   r  F where r  position vector, F  force (b) r Angular momentum J  rr  pr where r  position vector, p  linear momentum (c) Linear velocity v    r where r  position vector,   angular velocity (d) r Torque on dipole placed in electric field r  pr  E r where (e) r r p  dipole moment, E  Electric Field Tensor : A quantity that has different values in different directions is called tensor. Ex. Moment of Inertia In fact tensors are merely a generalisation of scalars and vectors ; a scalar is a zero rank tensor, and a vector is a first rank tensor. 15. Electric current is not a vector as it does not obey the law of vector addition.] 16. A unit vector has no unit. 17. To a vector only a vector of same type can be added and the resultant is a vector of the same type. 18. A scalar or a vector can never be divided by a vector.