1. Scalar quantities :Quantities which have only magnitude and no direction are called scalarquantities, e.g. mass, distance, time, speed, volume, density, pressure,work, energy, electric current, temperature, etc. Vector quantities :Quantities which have magnitude as well as direction and obey thetriangle law of vector addition or equivalently the parallelogram law ofvector addition are called vector quantities, e.g. position, displacement,velocity, force, acceleration, weight, momentum, impulse, electric field,magnetic field, current density, etc.

2. A vector quantity can be represented byan arrow. This arrow is called the vector.The length of the arrow represents themagnitude and the tip of the arrowrepresents the direction. If a car A runswith a velocity of 10 m/s towards east;and another car B runs with a velocity of20 m/s towards north-east. Thesevelocities can be represented by vectorsshown in the adjoining figure, takingeach unit of length on the arrow torepresent 5 m/s.NW ESVelocity ofcar AVelocity ofcar B 3. Two or more vectors are said to beequal if, and only if, they have thesame magnitude and same direction.In the adjoining figure, A, B and C areequal vectors.If the direction of a vector is reversed,the sign of the vector is reversed. Thisnew vector is called the negativevector of the original vector. Here,the vector D is the negative vector ofvectors A, B and C. Thus,A = B = C = -DABCD 4. Consider two vectors A and B.1. First vector A is drawn.2. Then starting from the arrow-headof A, the vector B is drawn.3. Now draw a vector R, starting fromthe initial point of A and ending atthe arrow-head of B. Vector R wouldbe the sum of A and B.R = A + BThe magnitude of A + B can bedetermined by measuring the length of Rand the direction can be expressed bymeasuring the angle between R and A(or B).ABR = A + BWe can start drawing from vector B also,instead of vector A, as shown below:-ABR = B + A 5. The vectors R and R obtained in the previous slide areparallel to each other and their lengths and directions aresame. Hence,R = R A + B = B + AThus, addition of vectors is commutative.This method of vector addition is called the method oftriangle of vectors. 6. There is another method ofadding two vectors, known asthe method of parallelogram ofvectors. According to thismethod, sum of two vectors Aand B is a vector R representedby the diagonal of aparallelogram whose adjacentsides are represented by vectorsA and B.AB 7. The magnitude of the sum of twovectors depends upon the anglebetween the vectors. In the adjoiningfigure, two vectors A and B are addedby changing the angle between them,keeping their magnitudes unchanged. Itis seen that the sum R of A and B ismaximum when A and B are parallel, i.e.,when the angle between them is 0. Themagnitude of R would be (A+B). Whenthe angle between A and B is 180, thenmagnitude of resultant vector R isminimum, equal to (A-B) if A is greater,or (B-A) if B is greater.ARBABRABRABRSince the minimum magnitude of A + B is(A-B), hence two vectors of differentmagnitudes cannot be added to get a zeroresultant. 8. If more than two vectors are to beadded, then we first determine the sumof any two vectors. The third vector isthen added to this sum and this methodis continued. Suppose we have to addfour vectors A, B, C and D as in theadjoining figure. Then we proceed asfollows:-R = (A + B) + C + DR = (E + C) + DR = F + DCA BDAE CBFDRThe sum of vectors in each case is thevector drawn to complete the polygonformed by the given vectors. Hence thismethod of addition of vectors is calledpolygon method. 9. ** The vectors need not be added in the order seen in the last slide.Vector C may be first added to vector A, then vector D and finallyvector B. R = A + C + D + BBut vector R and vector R are parallel, equal in length and are inthe same direction. R = Ror, A + B + C + D = A + C + D + BHence vector addition is associative.** If three or more vectors themselves complete a triangle or apolygon, then their sum-vector or resultant vector cannot bedrawn. It means that the sum of these vectors is zero. 10. (i) Triangle Law of Vector Addition:-This law states that if two vectors are represented in magnitude and directionby the two sides of a triangle taken in the same order, then their resultant isrepresented by the third side of the triangle taken in the opposite order.Let two vectors A and B be represented,both in magnitude and direction, by thesides OP and PQ of a triangle OPQ takenin the same order. Then the resultant Rwill be represented by the closing sideOQ taken in the opposite order.O PQRABE 11. To find the magnitude of resultant R, aperpendicular QE from Q on side OP producedis drawn. Let QPE = . Then, in right-angledOEQ, we have:-OQ = OE + QE= (OP + PE) + QE= OP + PE + 2.OP.PE + QENow, PE + QE = PQ OQ = OP + PQ + 2.OP.PEIn right-angled PEQ, we have cos = PE = PQ.cos OQ = OP + PQ + 2.OP.PQ.cos R = A + B + 2ABcos PEPQR = (A + B + 2ABcos ) 12. To find out the direction of the resultant,suppose the resultant R makes an angle withthe direction of vector A. Then,QEOEtan = =Now OP = A and PE = Bcos . To find QE, weconsider PEQ. We have:-sin = , or, QE = PQ sin = B sin . tan =QEOP + PEQEPQB sin A + B cos 13. (ii) Parallelogram Law of Vector Addition:-This law states that if two vectors are represented in magnitude and directionby the two adjacent sides of a parallelogram drawn from a point, then theirresultant is represented in magnitude and direction by the diagonal of theparallelogram drawn from the same point.Let two vectors A and B inclined to eachother at an angle be represented inmagnitude and direction, by the sides OPand OS of a parallelogram OPQS. Then,according to parallelogram law, the resultantof A and B is represented both in magnitudeand direction by the diagonal OQ of theparallelogram.BS Q ARO P E 14. As discussed in case of triangle law of vectoraddition, the magnitude and direction of theresultant R will be given by :-R = (A + B + 2ABcos )tan =B sin A + B cos 15. (i) When two vectors are in the same direction : Then, = 0 so thatcos = cos 0 = 1 and sin = sin 0 = 0. Then we have :-R = (A + B + 2AB.cos 0) = (A+B),and tan = = 0, i.e., = 0.B x 0A + BThus, the resultant R has a magnitude equal to the sum of the magnitudes of thevectors A and B and acts along the direction of A and B.(ii) When two vectors are at right angle to each other : Then, = 90 so thatcos 90 = 0 and sin 90 = 1. Then,R = (A + B + 2AB.cos 90) = (A + B),and tan = =B sin 90A + B cos 90BA 16. (iii) When two vectors are in opposite directions : Then, = 180, so thatcos 180 = -1 and sin 180 = 0. R = (A + B + 2AB.cos 180) = (A-B) = (A-B) or (B-A),B sin 180and tan = = 0, i.e., = 0 or 180.A + B cos 180Thus, the magnitude of the resultant vector is equal to the difference of themagnitudes of the two vectors and acts in the direction of the bigger vector.Note:- The magnitude of the resultant of two vectors is maximum when they are inthe same direction, and minimum when they are in opposite directions. 17. Further in a parallelogram, if one diagonal isthe sum of two adjacent sides, then the otherdiagonal is equal to its differences. In theadjoining figure,OQ = OP + PQPS = PQ + QSBut, QS = OPThus, PS = PQ OP 18. Suppose A and B are two vectors and thevector B is to be subtracted from vector A. Thesubtraction of B from A is same as addition ofB to A, i.e., A B = A + (B).Hence, to subtract vector B from A, first wereverse B to get B. Then the vector B isadded to vector A. For this, we first drawvector A and then starting from the arrow-headof A, we draw the vector B, and finallywe draw a vector R from the initial point of Ato the arrow-head of B. Thus, vector R is thesum of A and B, i.e., the difference A B :-R = A + (B) = A B.ABA-BA-BR = A B 19. On multiplying a vector A by a scalar or a number k, a vector R (say) isobtained :-R = k AThe magnitude of R is k times the magnitude of A and the direction of R issame as that of A. If k is a pure number having no unit, then the unit of Rwill be same as that of A. If a vector A is 5 cm long and directed towards east,then vector 2 A would be 10 cm long and directed towards east; and thevector -2 A would be 10 cm long but directed towards west. 20. If k is a physical quantity having a unit, then the unit of R will be obtained bymultiplying the units of k and A. In this case, the vector R will represent a newphysical quantity. For example, if we multiply a vector v (velocity) by a scalarm (mass) then their multiplication p (say) will represent a new vector quantitycalled momentum :-p = m vThe unit of mass m is kg and the unit of velocity v is m/s. Hence, the unit of pwill be kg.m/s. The direction of p will be same as that of v. 21. To describe the motion of an object in a plane, we use the concept ofposition and displacement vectors. For this, we select a point in the plane asorigin and describe the position of the object with respect to that origin.YPQr1r2O XSuppose, at an instant of time t1, the object is at a point P inthe X-Y plane of a cartesian coordinate system. Then avector OP drawn from origin O to the point P is called theposition vector of the object at time t1. It may be written asr1, where r1 is the distance of the point P from the origin O.If the object moves to a point Q at time t2, then OQ or r2 isthe position vector of the object at time t2, where r2 is thedistance of the point Q from the origin O. 22. YPQr1r2O XThe vector PQ drawn from the point P to the point Q is thedisplacement vector of the object during the interval t2 t1.The vector PQ is the vector difference OQ OP (sinceOP + PQ = OQ by triangle law of vector addition), i.e.,PQ = r2 r1Thus, the displacement vector is the difference between thefinal and the initial position vectors. 23. If two vectors A and B are equal, then their difference A B is defined as zero vectoror null vector and is denoted as 0.A B = 0, if A = B.Thus, zero vector is a vector of zero magnitude having no specific direction. Its initialand terminal points are coincident.Properties:- A + 0 = A n 0 = 0 0 A = 0 24. Though a zero vector does not quite fit in our description of a vector as ithas no specific direction, in this way it is considered as one of the non-propervectors. Still it is needed in vector algebra due to the followingreasons :- What is A B when A = B? What is A + B + C if these vectors form a closed figure? We know that with respect to origin in cartesian coordinate system, positionvector of a point P is OP, then whats the position vector of origin itself? What is the displacement vector of a stationary object? What is the acceleration vector of an object moving with a constant velocity?Answer to all these questions is a zero vector. 25. A vector whose magnitude is unity is called a unit vector.If A is a vector whose magnitude A 0, then A / A is a unit vector whose direction isthe direction of A. The unit vector in the direction of A is written as A. Thus,^ A^A = or, A = A A^AThus, any vector in the direction of unit vector may be written as the product of theunit vector and the scalar magnitude of that vector.Orthogonal Unit Vectors:- The unit vectorsalong the X-axis, Y-axis and Z-axis of the right-handedcartesian coordinate system are written^ ^ ^as i, j, and k respectively. These are calledorthogonal unit vectors.Y^O XZ^ji^k 26. The resolution of a vector is opposite to vector addition. If a vector is resolved intotwo vectors whose combined effect is the same as that of the given vector, then theresolved vectors are called the components of the given vector. If a vector isresolved into two vectors which are mutually perpendicular, then these vectors arecalled the rectangular components of the given vector.Let us suppose that a given vector A is to beresolved into two rectangular components. For this,taking the initial point of A as origin O, rectangularaxes OX and OY are drawn. Then perpendiculars aredropped on OX and OY from the arrow-head of A.These perpendiculars intersect OX and OY at P andQ respectively. Then the vectors Ax and Ay drawnfrom O to P and Q are the rectangular componentsof vector A. From rectangle OQRP, it is clear that thevector A is the sum of vectors Ax and Ay :-A = Ax + Ay.By measuring OP and OQ, the magnitudes of Axand Ay can be determined.XYQ ROPAAxAy 27. ^ ^Now let i and j be unit vectors along X and Y axesrespectively, and Ax and Ay the scalar magnitudes ofAx and Ay respectively. Then, we may write :-^ ^Ax = Ax i and Ay = Ay jThus, we have :-^ ^A = Ax i + Ay jThis is the equation for vector A in terms of itsrectangular components in a plane. If the vector Amakes an angle with the X-axis, then we have :-Ax = A cos and Ay = A sin From these equations, we have :-A = (Ax + Ay) = tan -1(Ay / Ax)Thus, if we know the magnitudes Ax and Ay of therectangular components of A, then from above twoequations, we can determine respectively themagnitude and direction of vector A. 28. The multiplication of two vector quantities cannot be done by simplealgebraic method. The product of two vectors may be a scalar as well as avector. For example, both force and displacement are vector quantities.Their product may be work as well as moment of force. Work is a scalarbut moment of force is a vector quantity.Vector quantities are represented by vectors. If the product of two vectors isa scalar quantity, then it is called scalar product; if the product is a vectorquantity then it is called vector product. If A and B are two vectors, thentheir scalar product is written as A B (read A dot B), and the vector productis written as A x B (read A cross B). Hence, the scalar product is also calleddot product and the vector product is also called cross product. 29. The scalar product of two vectors is defined as ascalar quantity equal to the product of theirmagnitudes and the cosine of the angle betweenthem. Thus, if is the angle between A and B, then,A B = A B cos ,where A and B are the magnitudes of A and B. Thequantity AB cos is a scalar quantity.ABNow, B cos is the component of vector B in the direction of A. Hence, thescalar product of two vectors is equal to the product of the magnitudeof one vector and the component of the second vector in the directionof the first vector. 30. (i) Power P is the rate of doing work. We know that :-Work W = Force F Displacement SWt t = F Hence, Power P = F vThus, power is the scalar product of force and velocity.(ii) The magnetic flux () linked with a plane is defined as scalar product ofuniform magnetic field B and vector area A of that plane :- = B AS 31. (i) The scalar product is commutative.A B = B A(ii) The scalar product is distributive.A (B + C) = A B + A C(iii) The scalar product of two mutually perpendicular vectors is zero.(iv) The scalar product of two parallel vectors is equal to the product of theirmagnitudes.(v) The scalar product of a vector with itself is equal to the square of themagnitude of the vector.^ ^ ^(vi) The scalar product of unit orthogonal vectors i, j, k have the followingrelations :-^ ^ ^ ^ ^ ^^ ^ ^ ^ ^ ^i j = j k = k i = 0i i = j j = k k = 1(vii) The scalar product of two vectors is equal to the sum of the products of theircorresponding x-, y-, z- components. A B = AxBx + AyBy + AzBz 32. The vector product of two vectors is defined as a vector having amagnitude equal to the product of the magnitudes of the two vectors andthe sine of the angle between them, and having the direction perpendicularto the plane containing the two vectors. Thus, if A and B be two vectors, thentheir vector product, written as A x B, is a vector C defined by :-^C = A x B = AB sin (A, B) n,where A and B are the magnitudes of A and B; (A, B) is the angle between them^and n is a unit vector perpendicular to the plane of A and B.^The direction of C (or n) is perpendicular to the plane containing A and B andits sense is decided by right-hand screw rule. 33. (i) Suppose there is a particle P of mass m whoseposition vector is r w.r.t the origin O of an inertialreference frame. Let p (= m v) be the linear momentumof the particle. Then, the angular momentum J of theparticle about the origin O is defined as the vectorproduct of r and p, i.e.,J = r x pIts scalar magnitude is J = r p sin ,where is the angle between r and p.(ii) The instantaneous linear velocity v of a particle isequal to the vector product of its angular velocity and its position vector r with reference to some origin,i.e., v = x rZYXOJr pP 34. (i) The vector product is not commutative, i.e.,A x B B x A(ii) The vector product is distributive, i.e.,A x (B + C) = A x B + A x C(iii) The magnitude of the vector product of two vectors mutually at right angles is equal to theproduct of the magnitudes of the vectors.(iv) The vector product of two parallel vectors is a null vector (or zero).(v) The vector product of a vector by itself is a null vector (zero), i.e.,A x A = 0^ ^ ^^ ^^ ^(vi) The vector product of unit orthogonal vectors i, j, k have the following relations :-^ ^^(a) i x j = j x i = k^^^j x k = k x j = i^ ^ ^^ ^k x i = i x k = j^ ^ ^ ^ ^ ^(b) i x i = j x j = k x k = 0(vii) The vector product of two vectors in terms of their x-, y- and z- components can beexpressed as a determinant. 35. Name:- Pankaj BhootraClass:- 11 CPhysics Project