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Mathematical methods, Q.1) Define scalars and vectors with examples., Ans :, i.Scalars:Physical, quantities, which, have, magnitude only and are specified completely by a, number are called Scalars., e.g.: length, mass, time volume, temperature,, speed, work, energy etc., ii.Vectors: Physical quantities which have, magnitude as well as direction are called vectors., eg.: Displacement, velocity, acceleration, force,, momentum, impulse etc., Q.2) Distinguish between Scalars and Vectors, Ans :, Scalars, Vectors, i) It has magnitude only. It has magnitude as, well as direction., ii) Scalars can be added Vectors are added or, or subtracted according subtracted, by, to the rules of algebra geometrical, (graphical)method, or vector algebra, iii) It has no proper It is represented by, symbol, symbol( )arrow., iv) The division of a scalar The division of a, by another scalar is vector by another, valid, vector is not valid, v) E.g. length, mass time, E.g. Displacement,, volume, etc., velocity,, acceleration, force,, etc., Q.3) Justify with an example mass is a scalar, quantity., Ans., i)Scalar quantities are the physical quantities, which have only magnitude and are specified, completely by a number and a unit., ii) When we say that the mass of a stone is 6 kg, it, indicates that the stone is 6 times heavier than, standard unit kilogram., iii)Therefore, number 6 is the magnitude.Unit, of mass is kilogram which gives us a complete, idea about the mass of the stone., iv)Thus mass is a scalar quantity., Q.4) How would you represent a vector, graphically and symbolically?, i)Graphical Representation :, Avector is graphically represented b y, adirected ine segment or an arrow. e.g., displacement of abody from P to Q is, Q., represented as P, 11th (Sci)) – Mathematical methods, , ii) Symbolical Representation :, Symbolically a vector is represented by a letter, with an arrow above it such as A . The magnitude, of the vector A is denoted as |A| or | A |or A., Q.5) Define the terms., i)Parallel vectors., ii)Antiparallel vectors., iii) Collinear vectors, iv) Negative vectors., Ans., i) Parallel vectors : Two or more vectors are, said to be parallel, if they act in the same, direction., a, b, , a| b ||c, , c, Parellel, ii) Anti-parallel vectors : Two vectors are said, to be anti parallel, if their directions are, opposite to one another., , a, b, Anti –Parallel vectors, iii) Collinear vectors : Two or more vectors are, said to be collinear if they are either parallel, or anti parallel or lie along the same line., , a and b are collinear vectors, iv) Negative vectors : A vector is said to be, negative of a given vector if its magnitude is, same as that of the given vector but its direction, is opposite Negative vectors are antiparallel, vectors., Infigure b = – a, Q.6) Define the terms., i) Zero (Null) vector. ii) Equal vectors., iii) Position vector., Ans. :, i) Zero vector(Null vector) : A vector which, haszero magnitude and having particular, direction is called zero vector., Example:Velocity vector of stationary particle,, acceleration vector of a body moving with, uniform velocity., , Page 1

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ii) Equal vectors : Two or more vectors having, thesame magnitude of their positions in space, are called equal vectors., , In the given figure | P | = | Q| = | R | = |S|, iii) Position vector :, A vector which gives the position of a particle, at a point with respect to the origin of chosen, co-ordinate system is called position vector., , R, R, , a, , b, , 3 4 7 units., , *, , Note : When vectors are not in the same direction, then they can added by using triangle law of, vectors additions., Q.10) What is triangle law of vectors addition?, Ans : Triangle law of vectors addition : Statement : If, two vectors of the same type are represented in, magnitude and direction, by the two sides of a, triangle taken in order, then their resultant is, represented in a magnitude and direction by the, third side of the triangle drawn from the starting, point of the first vector to the end point of the, second vector., , In the given figure OP represent position, vector of P with respectto O and OQ represent, position vector of Q with respect to O., Q. 7) Define., i) Resultant vector, ii)Composition of vectors., Ans :, i) Resultant vectors : The resultant of two or, more vectors is defined as that single vector,, which produces the same effect as produced by, all the vectors together., The process of, finding the resultant of two or more vectors is, called composition of vectors., Q.8) Whether, ether the resultant of two vectors of, unequal magnitude be zero?, Ans :The, The resultant of two vectors of differen, different, magnitude cannot give zero resultant., * Addition and subtraction of vectors., Q.9) What do you mean by addition of vectors?, Ans :, i) The addition of two or more vectors of same, type gives risee to a single vectors such that the, effect of this single vector is the same as the net, effect of the original vectors., ii) It is important to note that only the vectors of, the same type (physical quantity) can be added., iii)For example if two vectors a = 3 units and b, = 4 units are acting along the same line,, then they can be added as, 11th (Sci)) – Mathematical methods, , Let P and Q be the two vectors of same type, taken in same order as shown in figure. Resultant, vector will be given by third side taken in opposite, order., i)e. OA AB OB, R P Q, Q.11) Using triangle law of vector addition, explain, the process of adding two vectors which are, not lying in a straight line., Ans., i) Two vectors in magnitude and direction are, drawn in a plane as shown in fig. (a), Let these vectors are P and Q, , ii) Join the tail of Q to head of p in the given, direction. The resultant vector will be the line, which is obtained by joining tail of p to head of Q, as shown in fig. (b), , Page 2

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iii) If R is the resultant vector of P and Q then, using triangle law of vector addition, we have, R P Q, Note: The vector sum remains unchanged, even if, the order of vectors is interchanged., Q.12) Explain, how two vectors are subtracted, to findtheir resultant by using triangle, law of vector addition., Ans :, i) Let P and Q be the vectors in a plane as, shown in figure. (a), We have to find P Q, , iv), , From equation (1) and (2), P Q Q P, Hence additionof two vectors obey, commutative law., Q.14)Prove theassociative, ive property of vector, addition., Ans. Associative property of vector addition :, According to associative property, for three, vectors P Q R P Q R, , Proof., i), Let, AB Q BC R, to prove, P Q R P Q R, ii) To subtract Q from P , vector Q is reversed so, , ii), , Join OB and AC, In ΔOAB, OA AB OB (From triangle law of vectors, addition), P Q R1..........(1), In ΔOBC, OB BC OC (from triangle law of vectors, addition), R1 R S, [from equation (1)], P Q + R S..........(2), , iii), , In ΔABC, AB BC AC, Q R R 2 ..........(3), InΔOAC, OA AC OC, , that we get the vector– Q as shown in fig (b)., iii)The resultant vector, , R, , is obtained, , by, , joining tail of P to head of Q as shown in fig., (c)., iv) From triangle law of vector addition,, R P, Q P Q, Q.13)Prove that : addition of two vectors obey, commutative law., Ans : Proof :, i) Let two vectors P and Q are represented in, magnitude and direction by two sides OA and, AB respectively. To show that P Q Q P, iv), , P R2, , S, , from equation (3), P Q R S..........(4), ii) Complete a parallelogram OABC such that, OA CB P and AB OC Q Joint OB., iii) In Δ OCB, OC CB OB (By triangle law of, vector addition), P Q R......(1), In ΔOCB , OC CB OB (By triangle law of, vector addition), Q P R......(2), 11th (Sci)) – Mathematical methods, , Comparing equation (2) and (4), P Q R P Q R, Hence associative law is proved., Q.15) What is Law of polygon of vector addition., Ans :, Polygon law of vector addition: If a number of, vectors are represented in magnitude and, direction by the sides of an incomplete polygon, taken in order, then resultant is represented inin, Page 3

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magnitude and direction by the remaining side, of the polygon, directed from the starting point, of the first vector to the end point of last vector., , In ΔOAB, OB p1 p 2, In ΔOBC, OC p1 p 2 p3, In ΔOCD, OD p1 p 2 p3 p4, Q.16) State and prove parallelogram law of, vectors, addition, and, determine, magnitude and direction of resultant, vector., Ans : Parallelogram law of vector addition : If, two vectors of same type starting from the, same point, arerepresented in magnitude and, direction, on by the twoadjacent sides ofa, parallelogram, then their resultant, resultantvector is, given in magnitude and direction, by, thediagonal of the parallelogram starting, from the samepoint., , P Q R, Hence proved,, Magnitude of Resultant Vector, V, :, i) To find the magnitude of resultant vector, R OC , draw a perpendicular from C to meet, OA extended at D., and AC = OB = Q, CAD, BOA, ii) In right angle triangle ADC, cos =, AD = AC cos = Q cos ……….(3), and sin =, DC =AC sin = Q sin ……….(4), iii) Using Pythagoras theorem in right angle, triangle ODC, (OC)2 = (OD)2, + (DC)2, 2, 2, (OC) = (OA) + 2(OA).(AD), +(AD)2 +(DC)2…(5), iv) From, rom right angle triangle ADC, AD2 + DC2 =AC2 ……(6), v) From equation (5) and (6), we get, (OC)2 = (OA) 2 + 2(OA)(AD)+(AC)2 …. (7), vi) Using (3) and (7), we get, (OC)2 = (OA) 2 + (AC)2+ 2(OA)(AC) cos, R2 = P2+ Q2 + 2P.Q. cos, R=, 2, 67* ………. (8), Equation (8) gives the magnitude of resultant, vector R, * Direction of resultant vector R, Let R make an angle with P, In ΔODC, tan =, , Consider two vectors P and Q of the same, type, with their tails at the point ‘O’ and ‘ ’ in, the angle between P and Q ., Joint BC and AC to complete the parallelogram, OACB,with OA P and AB Q as the, adjacent sides. Wehave to prove that diagonal, OC R is, the resultant of sum the two given vectors, vectors., iii)By the triangle law of vector addition, we have, OA AC OC ……….(1), AC is parallel to OB, where AC = AC, , AC OB Q ……… (2), Substituting OA and OC in (1)we have,, 11th (Sci)) – Mathematical methods, , tan =, from equationss (3) and (4), we get, tan =, , # tan, , !", '1, , ), , …….. (9), *+,, cos, , 8……..(10), , Equation (10) represents direction of resultant, vector., Special cases :, Case 1 : When = 0° , i)e., e. P and Q are in the same, direction, then from equation (8), 1#, 2 cos 0°, :, #, 2, # 9, From equation (9), 2°, tan =, #0, 34 °, #0, , Page 4

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Case 2 : When # 90°i)e. when P and Q are, mutually perpendicular to each other,, then from equation (8), 1#, 2 cos 90, 90°, 1#, < cos 90° # 0A, From equation (9), =2°, 34 =2°, , #), , 8 # tan?@ ), , B@, 8, B2, , # tan?@ ) 8, Case 3 : When 0 = 180º , i)e. when P and Q are, in the opposite direction, then from, equation (8)., P2, , R, , Q2, , P2, , R, , 2PQ cos(180º ), , Q2, , 2PQ, , cos180º, , 1, , 1# 9 ' :, 1# ', From equation (9), tan, , 1, , Qsin180º, P Qcos180º, , 0, , # 0[ sin 180° = 0], Q.17) Define unit vector and give its physical, significance., Ans : Unit vector : A vector having unit, magnitude in a given direction is called a, unit vector., If P, C 0) then the, P, unit vector u p, P, P, * Significance of unit vector :, i) The unit vector gives the direction of a, given vector., ii) Unit vector along X, Y and Z direction of a, Cartesian coordinate is represented by i, j, and k ., Q.18) What are rectangular unit vectors? How, are they denoted?, Ans. :, i) The unit, vectors along the positive, directions of the axes of cartesian co, coordinate systems are called rectangular unit, vector., , ii) The unit vector along the positive direction of the, X-axis is denoted by i the unit vector along the, positive direction of the Y-axis, Y, is denoted by j and, the unit vector along the positive direction of the ZZ, axis is denoted by k ., iii) These unit vectors may not be located at the, origin. They can be translated anywhere in space, provided their directions with respect to the, respective axis remain same., iv)Three unit vectors are also called base vectors., Q.19) Define., i) Components of a vector., ii) Resolution of vectors., Ans., i)Components of a vector : The number of, vector whose combined effect is same as that of, the given vector are called components of the, given vector. The component of a vector in a, given direction gives the measure of the effect of, that vector in that direction., ii) Resolution of vectors :The process of, finding the components (or component vectors), of a givenvector is called resolution of vectors., Q.20) What are rectangular components of, vectors? Explain their uses., Ans :, i) Rectangular components of a vector: If, components, onents of a given vector are at right angles, an, to each other then they are called rectangular, components of that vector., ii) Rectangular components help us to find the, magnitude and direction of a vector when they, are resolvedd into two or more components., iii) Consider a vector R = OC originating from the, origin ‘O’ of a rectangular co-ordinate, co, system as, shows in figure., , Two dimensional rectangular components, , iv) Draw CA, OX and CB, OY., Let component of R along X–axis = R x, and components of R along Y–axis = R y, By parallelogram law of vectors, R R x R y AC OB R y or, 11th (Sci)) – Mathematical methods, , Page 5

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R iR x, , jR y, , Where, i and j are unit vectors along positive, direction X and Y axes respectively., v)If is angle made by R with X-axis then, D, , #, # E, D, Rx = R cos …….. (1), cos, , Ry= R sin, , between A and B ?, , Ans : If A B = A B then vectors A and B must, , ……… (2), , be at right angle to each other., , vi) Squaring and adding equation (1) and, R 2 sin 2, (2), we get R 2x R 2y R 2 cos 2, R 2 cos 2, , R 2x R 2y, R, , sin 2, , R2, , R 2x, , R 2y ............(3), , Equation (3) gives the magnitude of R ., viii) Similarly if R x , R y and R z are three, rectangular component of R along X, Y and Z, axes of a three dimensional rectangular, cartesian co-ordinate system then, R R x R y R z or, , R iR x jR y zR z, Magnitude of R is given by, R, , R 2x, , R 2y, , R 2z, , i, , 2, , 2, , j, , k, , 2, , 1, , Note :, If , F and G are angles made by R with Rx,, Ry, Rz then direction cosine of vector is given, bycos, , #, , DE, D, , , cosF #, , Ans :The diagonal of the parallelogram made by, two vectors as adjacent sides is not, passingthrough common point of two, vectors. This represents triangle law of, vector addition., Q.25) If A B A B then what can be the angle, , DH, D, , and cosG #, , DI, D, , Q.21) Whether it is possible to add two vectors, – representing physical quantities having, different dimensions?, Ans :, It is not possible to add two vectors, representing physical quantities having, different dimensions., Q.22) Is it possible to add two vectors using –, triangles law?, Ans : Yes is possible to add two vectors using, triangle law., Q.23) Whether the subtraction of given vectors a), is commutative or associative?, Ans :The subtraction of given vectors is neither, communicative nor associate, Q.24) The diagonal of the parallelogram made b), by two vectors as adjacent sides is not, passing through common point of two, vectors. What does it represent., 11th (Sci)) – Mathematical methods, , Q.26) What are (i) dimensions and (ii) units of, unit vectors?, Ans : Unit vector does not have any dimensions. Also, there is not unit for it.Where as unit vectors is, used to specifydirection only., Q.27) If the frame of reference is rotated or, displace, then what happens to the vector, and its components?, Ans : If the frame of reference is rotated or, displaced then a vector will not change since, it is independent of choice of frame of, reference. However the magnitude of, components will change with the co-ordinates, axes. This is because when the frame of, reference is rotated or displace, co-ordinates, axes changes., Q.28) Define scalar product of two vectors with, suitable examples., Ans : Scalar product of two vectors :, i), The scalar product of two non zero vectors is, defined as the product of the magnitude of the, two vectors and cosine of the angle between, then two vectors., ii), The dot sign is used betweenthe two vectors to, be multiplied therefore scalar product is also, called dot product., iii), The scalar product of two vectors P and Q is, given by, P.Q = QP cos, where, P = magnitude of P, Q = magnitude of Q, = angle between P and Q ., Examples of scalar product :, Work is a scalar product of force ( F ) and, displacement ( S ), W = F.S, Power ( P ) is a scalar product of force ( F ) and, velocity ( V ), P = F. V, , Page 6

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), , Magnetic flux (J) linked with a surface is a, scalar product of magnetic induction. ( B ) and, the area vector ( A ), K # B.A, Q.29) Give the geometrical meaning of dot, product of two vectors., Ans :, i) The scalar product of two vectors is, equivalent to the product of magnitude of one, vector with component of the other in the, direction of the first., ii) Let P and Q be two vectors lie at an angle, as shown in figure., , If P & Q are parallel,, P.Q = PQcos90° = 0, , v), , The converse is also true i)e. if the scalar, product of two non zero vectors.is zero, then, the vectors are perpendicular to each other., The scalar product of two parallel vectors is, equal to their product of magnitudes., If P & Q are parallel, = 0°& cos 0° = 1, P.Q = PQ cos 0° = PQ, , vi) Scalar product of rectangular unit vectors., , i.i | i | . | i | cos 0º 1, Similarly j.j 1, Thus, i.j, , This gives the geometrical meaning of, scalar product., iii) Fromdefinition of dot product., P.Q = PQ cos, = P(Q cos ), =P (component of Q in direction of P ), Similarly , Q . P = QP cos, = Q(P cos ), = Q(Components of P in direction Q ), Q.30) Discuss the main characteristics of, scalar product of two vectors., Ans : Characteristics of the scalar product, of two vectors :, i) The scalar product of two vectors obeys the, commutative law of multiplication., i)e. P.Q = PQ cos = QP cos, P.Q = Q.P, ii) The scalar product obeys the distributive law, ofmultiplication., i)e. P. Q R = P.Q P.R, iii) The scalar product of a vector with itself ((i)e., self dot product) is equal to the square of its, magnitude., P.P = PP cos0° = P2, The converse is also true i)e., e. if the scalar, product of two non zero vectors is zero, then, the vectors are perpendicular to each other., iv) The scalar product of two mutually, perpendicular vectors is equal to their, product of magnitudes., 11th (Sci)) – Mathematical methods, , = 0°& cos 0° = 1, , k.k = 1, , i . j cos90º 0, , i, , j,, , 90º, , Tabular form of dot product of unit vector is, given below., , i j k, i 1 0 0, j 0 1 0, k 0 0 1, Q.31) Derive an expression for scalar product of, vectors, in, termsof, their, scalar, components., Ans: Expression for scalar product of two, vectors :, i), , Let two vectors P and Q are represented in, magnitude by, P iPx, , jPy kPz, , Q iQ x, , ii), , jQ y, , kQ z, , Scalar product of P and Q is given by, , P.Q, , iPx, , jPy, , Px Q x i.i, , kPz . iQ x, , PxQ y i.j, Py Q y j.j, , Pz Q x k.i, , Pz Q y k.j, , j.k, , k.i, , kQz, , =, , Pz Qz i.k, , Py Q y j.i, , iii) Since, i.i, , jQ y, , Py Q z j.k, , Pz Qz k.k, , j.j k.k 1 and, j.i k.j i.k 0, , P.Q Px .Q x Py Qy Pz Qz, , Page 7

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Q.32) Define and explain the vector product, of two vectors, with suitable example., Ans : Statements :, i), The vector product of two vectors is a, thirdvector whose magnitude is equal to, the product of magnitude of the two, vectors and sine of the smaller angle, between the two vectors., ii), Vector product is also called cross product, of vectors because cross sign is used to, represent vector product., , * Explanation :, i) The vector product of two vectors P and Q ,, is third vector R and is written as, R P Q, The magnitude of R, , R = PQ sin , Where, , 0L LM, ii) The direction of the resultant vector R is, perpendicular to the plane formed by P and, Q , is a third vector R can be found by the, right hand screw rule., , iii) The instantaneous velocity v of a particle, is equal to the cross product of its angular, velocity, and its position vector r, , v, , r, , iv) The torque, , acting on a bar magnet freely, , suspended in a uniform magnetic induction, B is given by = M B where, M is the, magnetic dipole moment of the bar magnet., Q.33) State right handed screw rule to give, direction of cross product., Ans : Right handed screw rule :“Hold a right handed, screw with its axis perpendicular to the plane, containing vectors and the screw rotated from, first vector to second vector through a small, angle, the direction in which the screw tip, would advance is the direction of the vector, product of two vectors., vectors, , If R is the magnitude of resultant vector, v, product then R P Q PQ sin where n is a, unit vector in the direction of R ., , iii) If rotation is anticlockwise then resultant, vector is positive, whereas if rotation is, clockwise then resultant vector is negative., * Examples of vector product. :, i) Moment of a force or torque, is the vector, product of the position vector r and the force, F, , ii) The angular momentum L of the particle is the, vector product of the position vector r and the, linear momentum p of the particle., , Q.34) State the characteristics of the vector, product (cross product) of two vectors., Ans: Characteristics of the vector product(cross, product) :, i) The vector product of two vectors does not obey, the commutative law of multiplication., P Q Q P because P Q PQ sin, and Q P PQ sin(–, –θ) =– PQ sin, P Q, , Q P, , ii) The vector product obeys the distributibe law of, multiplication, i)e. A B C A B A C, iii)The vector product of a vector with itself(i)e., itself(, self, cross product) is equal to zero., P Q then P Q P P = PP sin0° = 0, , L r p, 11th (Sci)) – Mathematical methods, , Page 8

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iv) The vector product of two non zeroperpendicular, vectors is equal to the product of their, magnitudes. If P and Q are perpendicular, then, θ # 0° and sin 0° = 0, P Q = PQ sin θ° # 0, v) Vector product of two parallel of antiparallel, vectors is equal to zero., If P and Q are perpendicular, then ° # 0 and, sin 0° # 0, P Q PQ sin, 0, If P and Q are antiparallel, sin180°= 0, P Q PQ sin180º 0, , P Q, , P Q i k, Py Q y j j, , k, , i, , 0, , k, , j, , j, , k, , 0, , i, , k, , j, , i, , 0, , First, , Pz Q x k i, , Pz Q z k k, , j Px Qz Pz Q x, , iii) Determinant form of cross product of two vectors, , P and Q, P Q, , i, , j, , k, , Px, Q, , Py, Q, , Pz, Qz, , Show that magnitude of vector product of, two vectors is numerically equal to the area of, a parallelogram formed by the two vectors, , Ans : Suppose OACB is a parallelogram of adjacent, , j i, k, k j i, k i j, Cross product of unit vectors can also be, remembered by using table given below., Second, j, , j +, , k Px Q y PyQ x, , Q.36), , i, , Py Q z j i, , P Q i Py Q z Pz Q y, , vi) Vector product of rectangular unit vectors is, given by cycle rule. According to this rule, , i j k, j k i, b) If rotation is clockwise direction then, , Px Q y i, , Py Q x j i, , Pz Q y k j, , # 180° and, , a) Iff rotation is anticlockwise direction then, , Px Q x i j, , sides OA = P and OB Q, AOB, as shown in figure., We have to prove that area of parallelogram, , OACB = P Q, , vii) The vector product of two vectors can be, expressed in terms of their components., P Q, , i, , j, , k, , Px, Qx, , Py, Qx, , Pz, Qx, , Q.35) Derive an expression for cross product of, two vectors and express it in determinant, from., Ans : Expression for cross product of two vectors :, i), Let two vectors P and Q be represented in, magnitude and direction by, P iPx jPy kPz and Q iQ x jQ y kQ z, ii), , Cross product of vectors P and Q is given, by, P Q, , iPx, , jPy, , zPz, , iQ x, , jQ y, , zQ z, , Proof., In right angled ΔOBM, OBM, BM, h, sinθ =, OB OB, h = Ob sin θ, = Q sin, Now, Area of parallelogram, ram, OACB = Base B height., = OA B h = P(Q sin ) = PQ sin, PQ sin, , = P Q, , Area of parallelogram OACB = P Q, = magnitudes of the vector product., , 11th (Sci)) – Mathematical methods, , Page 9

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Note :If n is a unitvector perpendicular to the, , Master Problems, Scalars & vectors, , plane then P Q = PQ sin n, , 1) Two vectors P and Q have magnitudes 3 units, , P Q = PQ sin n, Q.37) Distinguish between scalar product (dot, product) and vector product (cross product), Ans:, No Scalar product, i) The magnitude of a scalar, product is equal to the, product of the magnitude, of the two vectors and the, cosine of the angle, between them, sin, ii), , P.Q = PQ, , It has no direction, , iii) It obeys the commutative, law of vector, multiplication., iv) It is zero if the two, vectors are mutually, perpendicular to each, other, v) The self dot product of a, vector is equal to the, square of its magnitude, , Vectors product, The magnitude of a, vector product is equal, to the product of the, magnitude of the two, vectors and sine of, small angle ( ) between, them., , P, , Q = PQ sin, , Its direction is, perpendicular to the, plane of the two vectors,, i)e. in the sense of, advancement of a righthanded screw, It does not obey the, commutative law of, vector multiplication., It is zero if the tow, vectors are parallel or, antiparallel to each other, The self cross product, of a vector is zero, , Q.38) Resultant of the product two vectors, always gives a vector true of false?, Explain., Ans: It is false. Resultant of the product of two, vectors does not always give vector. The dot, product of two vector is a scalar. For, example work., W = F.S and power , P = F.V etc., , and4 units respectively. R is the resultant of P, and Q . Find the magnitude and direction of R, when the angle between P and Q is 30°., 2) Two points A and B in space have the coordinates (2,–1, 3) and (4,2,5) respectively, find, vector AB., 3) What vector added to 2i 2 j k and 2i k, will give a vector along negative Y-axis?, 4) Find m if P, , i 3j 4k and mi 6 j 8k have the, , same direction, , 5) If A, , 3i 2 j 3k and B i j 2k . Find the, , angle between A, , 3i 2 j 3k and B i j 2k ., , Find the angle between A and B ., 6) The angular momentum L is a vector product of, positionvector r and linear momentum P ., If r 2i 3j 5k an d P 3i 4j 5k . Fi n al L., 7) Find the area of the parallelogram with adjacent, sides formed by P and Q , where P 2i 3j 4k, and Q, , 3i 2 j 2k expressed in meter., , 8) Two forces F1 and F2 of magnitude 5 N each inclined, to each other at 60°, act on a body. Find the, resultant force acting on the body., , 9) Aforce, , F 4i 6 j 3k, , acting, , on, , a, , particle, , produces a displacement S 2i 3j 5k where F is, expressed in newton and S in metre. Find the work, done by force, 10)If A, , 2i 7 j 3k and B 3i 2 j 5k find the, , component of A along B .., 11) In a certain co­ordinates system the co­ordinates of, two points P and Q are (2, 4, 4) and(–2,–3, 7), , respectively .Find PQ and its magnitude., 12)Find a if A 3i 2 j 4k and B ai 2 j k are, perpendicular to one another., 13) If P, , 2i 3j k and Q, , (i) P Q, , 2i 5j 2k , Find, , (ii) 3P 2Q ., , 14)Find the vector that should be added to sum of, 2i 5 j 3k and 4i 7 j 4k to give a unit vector, , along the X-axis?, , 11th (Sci)) – Mathematical methods, , Page 10

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15) Find unit vectors perpendiculars to the plane, of the vector. A i 2j k and B 2j k ., 16) Find the area of a triangle formed by, , A 3i 4 j 2k, and, B i j 2k, as, adjacent side measured in metre., 17) If A 6i 5 j 4k and B i 2 j 3k ,find the, resultant vector of the two vectors and its, magnitude ., 18) If A 5i 2 j 4k and B 2i j 2 k . Find a, vector which when added to the resultant of, A and B will give unit vector in z-axis., 19) A body experiences two forces P = 3 N and, Q = 2 N inclined to each other at an angle of, 30 °. Find the magnitude and direction of, resultant., 20) If R P Q and P Q R , find the angle, between P and Q ., 21) Find the unit vector in the direction of, resultant of P i 3j 5k and . Q i 2j 4k, 22) Find the work done when a force, F 2i 3j k N displaces a body through, , r 3i j 4k m., 23) Prove, , A, , that, , 2i 3j 4k, , and, , B 4i 4 j k are mutually perpendicular to, each other., 24) Calculate the angle between the vectors, P i 2j 4k and Q 4i j 2k ., 25) Unit vectors a and b inclined at an angle ,, prove that a b, 26) Evaluate (i) j, 27), , If A, , 2sin ., 2, i k (ii) 2i k, , i, , Formula : R, Ans : R, , 2) If P 6i j 2k and Q, , i 2 j 3k , Find the, , vector which when added to P Q will give the, vector 2i 3j k ., Formula : R, , 3) If A, , 7i 2j 4k, , 2i 3j 2k and B 4i 5j 3k , find the, , magnitudes of A, B and A, Formula : A, , A, , 2, x, , A, , 2, y, , B., A z2, , B2x B2y B2z, , Ans : B, , 4) If A i 2 j 3k and B 2i 3j 2k . Find, 2A 3B and 5A B ., Ans : 2A 2B, , 4i 5j 12k ;, , 5A B 7i 13j 13k, 5) A body is acted upon by two forces of mangitudes, F1# √2N and F2 = 3N which are inclined at 45° to, each other. Find the magnitude and direction of, the resultant force acting on the body., F12 F22 2F1F2 cos, , Formula : F, , F2 sin, F1 F2 cos, Ans :F= 4.123 N; a= 30°58`, 6) The resultant of two vectors each of equal, magnitude is 3P . What is the angle between the, tan, , Formula : a, , 2i j k ., and, , 90 unit., , P2, , Q2, , 2PQ cos, , 7) Fine a unit vector in the direction of 4i 8j k …, , 28) Find the area of parallelogram with adjacent, sides formed by P and Q where P i 2j 3k, , P 2i j k, , A B C, , Formula : R, Ans : # 60°, , Find (i) A B (ii) A B, , 29) If, , C i 2j k , find the magnitude of A B C ., , vectors?, , j, , 2i 2 j 3k and B i 3j k ., , and Q, , HOME WORK, 1) If A 3i j 4k , B 2i 4 j 3k and, , Q i j 2k, , and, , Ans : a, , A, A, , 4 8 1, i, j, k, 9 9 9, , 8) Find a unit vector in the direction of, , R i 2j k find P Q R P ., , A 3i 4j 12k ., 30), , Prove, , that, , the, , vectors, , 2i 3j k, , 6i 9 j 3k are called ., *****, , 11th (Sci)) – Mathematical methods, , and, Formula : A, , Ans : A, , A 2x, , A 2y, , A z2 ; A, , A, A, , 3, 4, 12, i, j, k, 13 13 13, Page 11

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9) Find the unit vector parallel to resultant of, vectors, , (ii) i 2 j, 16) A, , P 2i 6j 3k , Q 4i 3j k ., P Q; R R 2x, , Formula: R, , R 2y, , R 2z ; r, , R, , 6, 3, 4, i, j, k, 61, 61, 61, 10) Find the vector perpendicular to, , P Q, , 1, i, 3, , Ans : P Q, , i, Ax, Bx, , j, Ay, By, , k, i j, Az = 2 3, Bz, 4 2, , k, 6, 3, , Ans : A B 21i 30j 8k, 17) Find area of triangle formed by, , P Q, , 1, j, 3, , j k, , Evaluate A B ., , Formula : A B, , P 3i j 2k and Q 2i 2 j 4k, , 2i, , 2i 3j 6k and B 4i 2 j 3k, , R, , Ans : r, , Formula : Unit vector, , j k, , A, , 1, k, 3, , 3i 4 j 7k m and B, , i 2 j 3k m, , asadjacenent sides., , 11) Vector P, , 4i 5j 2k and Q ai 2 j k, are perpendicular to each other. Find ‘a’, Ans : a= –3, 12) A body constrained to move along the Z-axis, of co-ordinates system is subjected to a, i 2j 3k N . Calculate, the work done by this force in displacing the, , Formula : Area of triangle =, , A B, , i, Ax, Bx, , j, Ay, By, , 1, A B, 2, , k, Az ;, Az, , constant force F, , body through a distance of 4 k m along the, Z-axis., Formula : W F.r, , 18) If P i 2j 3k, Q, , j k,, , Formula : (i) A.B A x Bx, , A 2x, , Ay By Az Bz, , Formula : P Q, , Ans : P, , B, , B, , 2, y, , 2, z, , Py, Qy, , Qz ;, Qz, , Ans : A.B = 14.14, 15) Evaluate (i) 4i, , P Q, , 8i 2 j 4k, , 19) Show that, , Formula : P Q, , Formula : A.B = AB cos, , 2j i, , Px, Qx, , k, , P i 2j 3k and Q 2i 4 j 6k are parallel., , 14) Calculate A.B if two vectors of magnitude 5, and 4 inlicned at angle of 45°., , Ans :(i) 4i, , j, , B, , A.B, (iii) cos =, AB, Ans: # 180°, , (ii) i 2 j, , i, , A 2y A z2, 2, x, , (ii) B, , P Q, , ., , 2i 2 j 2k, , (i) A, , 2i 3j 4k , find P, , Ans : W = 12 J, , 13) What is the angle between A i, , B, , Ans :Are of triangle =13.9642 m2, , 2j i, , Ans :If P Q, , i, Px, Qx, , j, Py, Qy, , k, Pz, Qz, , 0 then P and Q are parallel to each, , other., , j k, 8j, , 11th (Sci)) – Mathematical methods, , Page 12

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Multiple Choice Question), , a) 90°, b) 60°, c) 30°, d) 0°, Two quantities of 5 and 12 units when added, gives a quantity 13 units. This quantity is, a), b), c), d), , 12) The vectors A and B are such that A B C, areA2 + B2 =C2. Angle, between positive, directions of A and B is, a) M/2 b) 0, c) M, d) 2M/3, 13) If A B C and magnitudes of A, B and C, and A2 + B2 = C2. Angle between positive, directions of A and B is., a) sin–1(3/4), b)cos–1(4/5), c)tan–1(5/3), d)cos–1(3/5), 14) Out of the following set of forces, the resultant, of which set of forces can never be zero?, a) 15, 15, 15, b) 15, 30, 60, c) 15, 15, 30, d) 15, 30, 30, , If a b, , 15) The expression, , 1., , Which of the following is a vector?, a) speed, b) displacement, c) mass, d) time, , 2., , The angle between the vectors a, , b 6i 4j, 3., , 4., , 2i 3j and, , is, , a b and a and b are non zero, , vectors, then, a) a and b, , b) a, , b, , c) a and b are zero parallel vectors d) a isU b, 5), , The equation a a a is, a) meaningless, b) always true, c) many be possible for limited values of a, , 6), , If R, , P Q , and, , between P and Q is, a) 30°, b) 60°, 7), , 8), , 9), , 10), , 11), , R2 = P2 +Q2, then angle, c), , 90°, , d) 180°, , i, , j is a, , a) unit vector, c) vectors of magnitude, , b) null vector, d)scalar, , 16) What is the angle between i j k and i ?, a) 0°, b)M/6, c)M/3, d) None of these above, 17) P Q is a until vector along X-axis?, If P, , i j k , then Q is, , a) i j k, , d) true only when a = 0, , @, , √, , b) j k, , c) i j k, d) j k, 18) The magnitude of scalar product of the vectors, A 2i 5k and B 3i 4k is, a) 20, b) 22, c) 26, d) 29, , The resultant of two vectors of magnitude P is, 19) If A i 2j 3k and B 3i 2 j k then the, area of parallelogram formed from these vectors, also P . They act at an angle, as the adjacent sides willbe, a) 60°, b) 90°, c) 120°, d) 180°, a) 2√3 square units, b) 4√3 square units, Given that R P Q .Which of the following, c) 6√3 square units, d) 8√3 square units, 20) Three vectors A, B and C satisfy the relation, relations is necessarily valid?, a) P < Q, b) P > Q c) P=Q d) None of these, A.B 0 and A.C 0 then A is parallel to, The minimum number of numerically equal, a) B, b) C, c) B C, d) B.C, vectors whose vector sum can be zero is?, 21) What vector must be added to the sum of two, a) 4, b) 3, c) 2, d) 1, vectors 2i j 3k and 3i 2j 2k so that the, A force of 60N acting perpendicular to a force, resultant is a unit vector along Z-axis?, 80N, magnitude of resultant force is., a) 5j k b) 5i 3j c) 3j 5k, d) 3j 2k, a) 20 N, b) 70N c) 100 N d)140 N, 22) The maximum value of magnitude of A B is, A river is flowering at the rate of 60 km h–1., A man swims across it with a velocity of 9 km, a) A – B b) A, c) A+B d) A B, h–1. The resultant velocity of the man will be, 23) The magnitudes of the X and Y components of, a) √15 km h–1, b)√45 km h–1, A are 7 and 6. Also the magnitudes of the X and Y, d) 9225: km h–1, c) √117 km h–1, components of A B are 11 and 9 respectively., What is the magnitude of B ., 11th (Sci)) – Mathematical methods, , Page 13

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a) 4, 24) If A, , b) 6, , c) 8, , d) 9, , 5i 2 j 3k and B 2i j 2k then, , component of B along A is, , a), , 28, 38, , b), , 28, 38, , c), , 28, 48, , d), , 14, 38, , 25) If A B A B then component of B must, be, , 26), , 27), , 28), , 29), , 30), , 31), , 32), , a) zero vector, b) unit vector, c) Non zero vector, d) equal to A, Choose the WRONG statement, a) The division of vector by scalar is valid, b) The multiplication of vector by scalar is, valid, c) The multiplication of vector by another, vector is valid by using vector algebra, d) The division of a vector by another vector, is valid by using vector algebra., A person moves from a point S and walks, along the path which is square. Of each side, 50 m. Each side runs east south, then west, and finallynorth. Then the total displacement, covered is, d) zero, a) 200 m b) 100 m, c) 50√2, Walking on the road is an example of, a) vectors, b) scalars, c) resolution of vectors d) null vector, Which of the following is a scalar?, a) Electric field, b) Angular momentum, c) Angular frequency d) Torque, Which of the of following is a vector?, a) Pressure, b) Gravitational potential, c) Angle, d) Current density, The resultant of two forces of 3 N and 4 N is 5 N the, angle between the forces is, a) 30°, b) 60°, c) 90°, d) 120°, Let the angle between two non­zero vectors, A and B be 120° and it's resultant be C, a) C must be equal to A B, , 34) What is the maximum number of components, into which a force can be resolved?, a) two, b) Three, c) Four, d)Any number, 35) The unit vector along i j is, , i j, 2, 2, 36) Two physical quantities, one of which is vector and, other is scalar, having same dimensions are, a) Work and energy Work and torque, b) Work and torque, c) Pressure and power, d) Impulse and momentum, 37) If vectors P and Q are perpendicular to each, other, then, a) P.Q 0 b) P Q 0 c) P Q d) P Q 0, a) K, , b) i j, , c), , i j, , d), , 38) If A, B and C are aren a zero vectors and, , 39), , 40), , 41), , A.B = 0 and B.C = 0 then magnitude of A.C is, a) A+C, b) AC, c) AB, d) BC, The negative vector of a given vector is vector of, a) same magnitude and same direction, b) different magnitude and same direction, c) different magnitude and different direction, d) same magnitude but directed opposite to that of, the given vector, The two vectors have same magnitude and same, direction, then they are said to be, a) null vector, b) opposite vector, c) equal vector, d) unit vectors, Vectors subtraction,, a) does obey commutative law, b) does obey associative law, c) does obey commutative and associative law, d) 5 does not obey commutative and associative,, , b) C must be less than A B, c) C must be greater than A B, d) C must be zero, 33) If n is the unit vector in the direction of A, them, , a) n, , A, , b) n, , AA, , d) n, , n A, , P 2 Q2, , 2PQsin, , d), , P 2 Q2, , 2PQ cos, , A, A, c) n, , A, , 11th (Sci)) – Mathematical methods, , Page 14