Chapter-6-VECTOR -01-THEORY-I

6.1 Introduction. Vectors represent one of the most important mathematical systems, which is used to handle certain types of problems in Geometry, Mechanics and other branches of Applied Mathematics, Physics and Engineering. Scalar and vector quantities : Physical quantities are divided into two categories – scalar quantities and vector quantities. Those quantities which have only magnitude and which are not related to any fixed direction in space are called scalar quantities, or briefly scalars. Examples of scalars are mass, volume, density, work, temperature etc. A scalar quantity is represented by a real number along with a suitable unit. Second kind of quantities are those which have both magnitude and direction. Such quantities are called vectors. Displacement, velocity, acceleration, momentum, weight, force etc. are examples of vector quantities. 6.2 Representation of Vectors. Geometrically a vector is represented by a line segment. For example, . Here A is called the initial point and B, the terminal point or tip. Magnitude or modulus of is expressed as . Note :  The magnitude of a vector is always a non-negative real number.  Every vector has the following three characteristics: Length : The length of will be denoted by or AB. Support : The line of unlimited length of which AB is a segment is called the support of the vector . Sense : The sense of is from A to B and that of is from B to A. Thus, the sense of a directed line segment is from its initial point to the terminal point. 6.3 Types of Vector. (1) Zero or null vector : A vector whose magnitude is zero is called zero or null vector and it is represented by . The initial and terminal points of the directed line segment representing zero vector are coincident and its direction is arbitrary. (2) Unit vector : A vector whose modulus is unity, is called a unit vector. The unit vector in the direction of a vector is denoted by , read as “a cap”. Thus, . Note :  Unit vectors parallel to x-axis, y-axis and z-axis are denoted by i, j and k respectively.  Two unit vectors may not be equal unless they have the same direction. (3) Like and unlike vectors : Vectors are said to be like when they have the same sense of direction and unlike when they have opposite directions. (4) Collinear or parallel vectors : Vectors having the same or parallel supports are called collinear vectors. (5) Co-initial vectors : Vectors having the same initial point are called co-initial vectors. (6) Co-planar vectors : A system of vectors is said to be coplanar, if their supports are parallel to the same plane. Note :  Two vectors having the same initial point are always coplanar but such three or more vectors may or may not be coplanar. (7) Coterminous vectors : Vectors having the same terminal point are called coterminous vectors. (8) Negative of a vector : The vector which has the same magnitude as the vector but opposite direction, is called the negative of and is denoted by . Thus, if , then . (9) Reciprocal of a vector : A vector having the same direction as that of a given vector but magnitude equal to the reciprocal of the given vector is known as the reciprocal of and is denoted by . Thus, if Note :  A unit vector is self reciprocal. (10) Localized and free vectors : A vector which is drawn parallel to a given vector through a specified point in space is called a localized vector. For example, a force acting on a rigid body is a localized vector as its effect depends on the line of action of the force. If the value of a vector depends only on its length and direction and is independent of its position in the space, it is called a free vector. (11) Position vectors : The vector which represents the position of the point A with respect to a fixed point O (called origin) is called position vector of the point A. If (x, y, z) are co-ordinates of the point A, then . (12) Equality of vectors : Two vectors and are said to be equal, if (i) (ii) They have the same or parallel support and (iii) The same sense. 6.4 Rectangular resolution of a Vector in Two and Three dimensional systems. (1) Any vector can be expressed as a linear combination of two unit vectors and at right angle i.e., The vector and are called the perpendicular component vectors of . The scalars x and y are called the components or resolved parts of in the directions of x-axis and y-axis respectively and the ordered pair (x, y) is known as co-ordinates of point whose position vector is . Also the magnitude of and if be the inclination of with the x-axis, then (2) If the coordinates of P are (x, y, z) then the position vector of can be written as . The vectors and are called the right angled components of . The scalars are called the components or resolved parts of in the directions of x-axis, y-axis and z-axis respectively and ordered triplet (x, y, z) is known as coordinates of P whose position vector is . Also the magnitude or modulus of Direction cosines of are the cosines of angles that the vector makes with the positive direction of x, y and z-axes. , and Clearly, . Here , and are the unit vectors along respectively. Example: 1 If is a non-zero vector of modulus a and m is a non-zero scalar, then is a unit vector if (a) (b) (c) (d) Solution: (c) As is a unit vector,    Example: 2 For a non-zero vector , the set of real numbers, satisfying consists of all x such that (a) (b) (c) (d) Solution: (b) We have,     . Example: 3 The direction cosines of the vector are (a) (b) (c) (d) Solution: (b) ; Hence, direction cosines are i.e., . 6.5 Properties of Vectors. (1) Addition of vectors (i) Triangle law of addition : If two vectors are represented by two consecutive sides of a triangle then their sum is represented by the third side of the triangle, but in opposite direction. This is known as the triangle law of addition of vectors. Thus, if and then i.e., . (ii) Parallelogram law of addition : If two vectors are represented by two adjacent sides of a parallelogram, then their sum is represented by the diagonal of the parallelogram whose initial point is the same as the initial point of the given vectors. This is known as parallelogram law of addition of vectors. Thus, if and Then i.e., , where OC is a diagonal of the parallelogram OABC. (iii) Addition in component form : If the vectors are defined in terms of , j and k, i.e., if and , then their sum is defined as . Properties of vector addition : Vector addition has the following properties. (a) Binary operation : The sum of two vectors is always a vector. (b) Commutativity : For any two vectors and , (c) Associativity : For any three vectors and , (d) Identity : Zero vector is the identity for addition. For any vector , (e) Additive inverse : For every vector its negative vector exists such that i.e., is the additive inverse of the vector a. (2) Subtraction of vectors : If and are two vectors, then their subtraction is defined as where is the negative of having magnitude equal to that of and direction opposite to . If and Then . Properties of vector subtraction (i) (ii) (iii) Since any one side of a triangle is less than the sum and greater than the difference of the other two sides, so for any two vectors a and b, we have (a) (b) (c) (d) (3) Multiplication of a vector by a scalar : If is a vector and m is a scalar (i.e., a real number) then is a vector whose magnitude is m times that of and whose direction is the same as that of , if m is positive and opposite to that of , if m is negative. Magnitude of  m (magnitude of ) = Again if then Properties of Multiplication of vectors by a scalar : The following are properties of multiplication of vectors by scalars, for vectors and scalars m, n (i) (ii) (iii) (iv) (v) (4) Resultant of two forces where = P, , Deduction : When , i.e., P = Q, ; Hence, the angular bisector of two unit vectors and is along the vector sum . Important Tips  The internal bisector of the angle between any two vectors is along the vector sum of the corresponding unit vectors.  The external bisector of the angle between two vectors is along the vector difference of the corresponding unit vectors. Example: 4 If is a regular hexagon, then (a) (b) (c) (d) Solution: (d) We have = = = = Example: 5 The unit vector parallel to the resultant vector of and is (a) (b) (c) (d) Solution: (a) Resultant vector = Unit vector parallel to = = Example: 6 If the sum of two vectors is a unit vector, then the magnitude of their difference is (a) (b) (c) (d) 1 Solution: (b) Let , and      . Example: 7 The length of longer diagonal of the parallelogram constructed on 5a + 2b and a – 3b, it is given that and angle between a and b is , is (a) 15 (b) (c) (d) Solution: (c) Length of the two diagonals will be and  , Thus, = = 15. = = . Length of the longer diagonal = Example: 8 The sum of two forces is and resultant whose direction is at right angles to the smaller force is . The magnitude of the two forces are [AIEEE 2002] (a) 13, 5 (b) 12, 6 (c) 14, 4 (d) 11, 7 Solution: (a) We have, ;   Now,    and  , Magnitude of two forces are 5N, 13N. Example: 9 The vector , directed along the internal bisector of the angle between the vectors and with , is (a) (b) (c) (d) Solution: (a) Let = and Now required vector = =   = 6.6 Position Vector . If a point O is fixed as the origin in space (or plane) and P is any point, then is called the position vector of P with respect to O. If we say that P is the point , then we mean that the position vector of P is with respect to some origin O. (1) in terms of the position vectors of points A and B : If and are position vectors of points A and B respectively. Then, In , we have  = (Position vector of B) – (Position vector of A)  = (Position vector of head) – (Position vector of tail) (2) Position vector of a dividing point (i) Internal division : Let A and B be two points with position vectors and respectively, and let C be a point dividing AB internally in the ratio m : n. Then the position vector of C is given by (ii) External division : Let A and B be two points with position vectors and respectively and let C be a point dividing AB externally in the ratio m : n. Then the position vector of C is given by Important Tips  Position vector of the mid point of AB is  If are position vectors of vertices of a triangle, then position vector of its centroid is  If are position vectors of vertices of a tetrahedron, then position vector of its centroid is . Example: 10 If position vector of a point A is and divides AB in the ratio 2 : 3, then the position vector of B is (a) (b) (c) (d) Solution: (c) Let position vector of B is . The point divides AB in 2 : 3.  Example: 11 Let , , be distinct real numbers. The points with position vectors , , (a) Are collinear (b) Form an equilateral triangle (c) Form a scalene triangle (d) Form a right angled triangle Solution: (b) ABC is an equilateral triangle. Example: 12 The position vectors of the vertices A, B, C of a triangle are , and respectively. The length of the bisector AD of the angle BAC where D is on the segment BC, is (a) (b) (c) (d) None of these Solution: (a) = = = . BD : DC = AB : AC = . Position vector of D = = position vector of D – Position vector of A = = . Example: 13 The median AD of the triangle ABC is bisected at E, BE meets AC in F. Then AF : AC = (a) 3/4 (b) 1/3 (c) 1/2 (d) 1/4 Solution: (b) Let position vector of A with respect to B is and that of C w.r.t. B is . Position vector of D w.r.t. B = Position vector of E = .....(i) Let AF : FC = and Position vector of F = Now, position vector of E = .......(ii). From (i) and (ii) ,  and  , . 6.7 Linear Combination of Vectors. A vector is said to be a linear combination of vectors etc, if there exist scalars x, y, z etc., such that Examples : Vectors are linear combinations of the vectors . (1) Collinear and Non-collinear vectors : Let and be two collinear vectors and let be the unit vector in the direction of . Then the unit vector in the direction of is or according as and are like or unlike parallel vectors. Now, and .   , where . Thus, if are collinear vectors, then or for some scalar . (2) Relation between two parallel vectors (i) If and be two parallel vectors, then there exists a scalar k such that . i.e., there exist two non-zero scalar quantities x and y so that . If and be two non-zero, non-parallel vectors then  and . Obviously  (ii) If and then from the property of parallel vectors, we have (3) Test of collinearity of three points : Three points with position vectors are collinear iff there exist scalars x, y, z not all zero such that , where . If , and , then the points with position vector will be collinear iff . (4) Test of coplanarity of three vectors : Let and two given non-zero non-collinear vectors. Then any vectors coplanar with and can be uniquely expressed as for some scalars x and y. (5) Test of coplanarity of Four points : Four points with position vectors are coplanar iff there exist scalars x, y, z, u not all zero such that , where . Four points with position vectors , , , will be coplanar, iff 6.8 Linear Independence and Dependence of Vectors. (1) Linearly independent vectors : A set of non-zero vectors is said to be linearly independent, if . (2) Linearly dependent vectors : A set of vectors is said to be linearly dependent if there exist scalars not all zero such that Three vectors , and will be linearly dependent vectors iff . Properties of linearly independent and dependent vectors (i) Two non-zero, non-collinear vectors are linearly independent. (ii) Any two collinear vectors are linearly dependent. (iii) Any three non-coplanar vectors are linearly independent. (iv) Any three coplanar vectors are linearly dependent. (v) Any four vectors in 3-dimensional space are linearly dependent. Example: 14 The points with position vectors , are collinear, if a = (a) – 40 (b) 40 (c) 20 (d) None of these Solution: (a) As the three points are collinear, such that x, y, z are not all zero and .  and  , and For non-trivial solution,  Trick : If A, B, C are given points, then  On comparing,  and  . Example: 15 If the position vectors of A, B, C, D are and respectively and , then will be (a) – 8 (b) – 6 (c) 8 (d) 6 Solution: (b) ; ;   . Example: 16 Let and be three non-zero vectors such that no two of these are collinear. If the vector is collinear with and is collinear with ( being some non-zero scalar) then equals (a) 0 (b) (c) (d) Solution: (a) As and are collinear ......(i) Again is collinear with = .....(ii) Now, = = .....(iii) Also, = = ......(iv) From (iii) and (iv), But and are non-zero , non-collinear vectors, . Hence, . Example: 17 If the vectors and are coplanar, then m is (a) 38 (b) 0 (c) 10 (d) – 10 Solution: (c) As the three vectors are coplanar, one will be a linear combination of the other two.  .....(i) .....(ii) .....(iii) From (i) and (ii), ; From (iii), . Trick : Vectors , and are coplanar.      . Example: 18 The value of for which the four points are coplanar (a) 8 (b) 0 (c) – 2 (d) 6 Solution: (c) The given four points are coplanar and , where , w are not all zero.  + and  , , and For non-trivial solution,  , Operating   . Example: 19 If and are linearly dependent vectors and , then (a) (b) (c) (d) Solution: (d) The given vectors are linearly dependent hence, there exist scalars not all zero, such that i.e., , i.e.,  , , For non-trivial solution,    ; Trick :  are linearly dependent, hence  .  . 6.9 Product of Two Vectors. Product of two vectors is processed by two methods. When the product of two vectors results is a scalar quantity, then it is called scalar product. It is also known as dot product because we are putting a dot (.) between two vectors. When the product of two vectors results is a vector quantity then this product is called vector product. It is also known as cross product because we are putting a cross (×) between two vectors. (1) Scalar or Dot product of two vectors : If and are two non-zero vectors and be the angle between them, then their scalar product (or dot product) is denoted by and is defined as the scalar , where are modulii of and respectively and . Important Tips     angle between and is acute.   angle between and is obtuse.  The dot product of a zero and non-zero vector is a scalar zero. (i) Geometrical Interpretation of scalar product : Let and be two vectors represented by and respectively. Let be the angle between and . Draw and . From and , we have and . Here and are known as projection of on a and on respectively. Now = = = .....(i) Again, = a.b = (Magnitude of b) (Projection of on ) .....(ii) Thus geometrically interpreted, the scalar product of two vectors is the product of modulus of either vector and the projection of the other in its direction. (ii) Angle between two vectors : If be two vectors inclined at an angle , then,   If and ; (2) Properties of scalar product (i) Commutativity : The scalar product of two vector is commutative i.e., . (ii) Distributivity of scalar product over vector addition: The scalar product of vectors is distributive over vector addition i.e., (a) (Left distributivity) (b) (Right distributivity) (iii) Let and be two non-zero vectors . As are mutually perpendicular unit vectors along the co-ordinate axes, therefore ; . (iv) For any vector . As are unit vectors along the co-ordinate axes, therefore , and (v) If m is a scalar and be any two vectors, then (vi) If m, n are scalars and be two vectors, then (vii) For any vectors and , we have (a) (b) (viii) For any two vectors and , we have (a) (b) (c) (d)  (e) (f) (3) Scalar product in terms of components.: If and , then, . Thus, scalar product of two vectors is equal to the sum of the products of their corresponding components. In particular, . Example: 20 (a) (b) (c) (d) Solution: (a) Let , . Example: 21 If then a value of for which is perpendicular to is (a) 9/16 (b) 3/4 (c) 3/2 (d) 4/3 Solution: (b) is perpendicular to    Example: 22 A unit vector in the plane of the vectors , and orthogonal to is (a) (b) (c) (d) Solution: (b) Let a unit vector in the plane of and be = As is unit vector, we have = +  .....(i) As is orthogonal to , we get   From (i), we get   . Thus Example: 23 If be the angle between the vectors and , then (a) (b) (c) (d) Solution: (a) Angle between and is given by, = Example: 24 Let and be vectors with magnitudes 3, 4 and 5 respectively and = 0, then the values of is (a) 47 (b) 25 (c) 50 (d) – 25 Solution: (d) We observe, = = = – 9 Trick: Squaring both the sides    . Example: 25 The vectors and make an obtuse angle whereas the angle between b and k is acute and less than , then domain of is (a) (b) (c) (d) Null set Solution: (d) As angle between and is obtuse,     ......(i) Angle between and is acute and less than .         or ……(ii) From (i) and (ii), . Domain of is null set. Example: 26 In cartesian co-ordinates the point A is where on the curve . The tangent at A cuts the x-axis at B. The value of the dot product is (a) (b) (c) 140 (d) 12 Solution: (b) Given curve is ......(i) when , ; From (i), Equation of tangent at A is   This tangent cuts x-axis (i.e., ) at ; = = = . Example: 27 If three non-zero vectors are , and . If c is the unit vector perpendicular to the vectors a and b and the angle between a and b is , then is equal to (a) 0 (b) (c) 1 (d) Solution: (d) As is the unit vector perpendicular to and , we have Now, = = = (4) Components of a vector along and perpendicular to another vector : If and be two vectors represented by and . Let be the angle between and . Draw . In , we have Thus, and are components of along and perpendicular to respectively. Now, = = =  Thus, the components of along and perpendicular to are and respectively. Example: 28 The projection of on is (a) (b) (c) (d) Solution: (b) Projection of on = = = Example: 29 Let be such that . If the projection along is equal to that of along and , are perpendicular to each other then equals (a) 14 (b) (c) (d) 2 Solution: (c) Without loss of generality, we can assume and . Let ,  .....(i) Projection of along = Projection of along     Now, = = = . Example: 30 Let , and let and be component vectors of parallel and perpendicular to a. If , then (a) (b) (c) (d) None of these Solution: (b) = = Clearly, i.e., is parallel to ; is to . Example: 31 A vector a has components 2p and 1 with respect to a rectangular cartesian system. The system is rotated through a certain angle about the origin in the anti-clockwise sense. If a has components and 1 with respect to the new system, then (a) (b) or (c) or (d) or Solution: (b) Without loss of generality, we can write .....(i) Now, From (i),   ....(ii) and ....(iii) Squaring and adding,    , . (5) Work done by a force : A force acting on a particle is said to do work if the particle is displaced in a direction which is not perpendicular to the force. The work done by a force is a scalar quantity and its measure is equal to the product of the magnitude of the force and the resolved part of the displacement in the direction of the force. If a particle be placed at O and a force represented by be acting on the particle at O. Due to the application of force the particle is displaced in the direction of . Let be the displacement. Then the component of in the direction of the force is cos . Work done = , where Or Work done = (Force) . (Displacement) If a number of forces are acting on a particle, then the sum of the works done by the separate forces is equal to the work done by the resultant force. Example: 32 A particle is acted upon by constant forces and which displace it from a point to the point . The work done in standard units by the force is given by (a) 15 (b) 30 (c) 25 (d) 40 Solution: (d) Total force = = Displacement = = Work done = = = . Example: 33 A groove is in the form of a broken line ABC and the position vectors of the three points are respectively , and . A force of magnitude acts on a particle of unit mass kept at the point A and moves it along the groove to the point C. If the line of action of the force is parallel to the vector all along, the number of units of work done by the force is (a) (b) (c) (d) Solution: (c) = = Displacement position vector of C – Position vector of A = = Work done by the force = = . 6.10 Vector or Cross product of Two Vectors. Let be two non-zero, non-parallel vectors. Then the vector product , in that order, is defined as a vector whose magnitude is where is the angle between and whose direction is perpendicular to the plane of and in such a way that and this direction constitute a right handed system. In other words, where is the angle between and , is a unit vector perpendicular to the plane of and such that form a right handed system. (1) Geometrical interpretation of vector product : If be two non-zero, non-parallel vectors represented by and respectively and let be the angle between them. Complete the parallelogram OACB. Draw . In  ......(i) Now, = = = = Vector area of the parallelogram OACB Thus, is a vector whose magnitude is equal to the area of the parallelogram having and as its adjacent sides and whose direction is perpendicular to the plane of and such that form a right handed system. Hence represents the vector area of the parallelogram having adjacent sides along and . Thus, area of parallelogram OACB = . Also, area of area of parallelogram OACB = (2) Properties of vector product (i) Vector product is not commutative i.e., if and are any two vectors, then , however, (ii) If are two vectors and m is a scalar, then (iii) If are two vectors and m, n are scalars, then (iv) Distributivity of vector product over vector addition. Let be any three vectors. Then (a) (Left distributivity) (b) (Right distributivity) (v) For any three vectors we have (vi) The vector product of two non-zero vectors is zero vector iff they are parallel (Collinear) i.e., are non-zero vectors. It follows from the above property that for every non-zero vector , which in turn implies that (vii) Vector product of orthonormal triad of unit vectors i, j, k using the definition of the vector product, we obtain , (viii) Lagrange's identity: If a, b are any two vector then or (3) Vector product in terms of components : If and . Then, . (4) Angle between two vectors : If is the angle between and , then Expression for : If , and be angle between and , then (5) (i) Right handed system of vectors : Three mutually perpendicular vectors form a right handed system of vector iff , Example: The unit vectors , k form a right-handed system, (ii) Left handed system of vectors : The vectors , mutually perpendicular to one another form a left handed system of vector iff (6) Vector normal to the plane of two given vectors : If be two non-zero, nonparallel vectors and let be the angle between them. where is a unit vector to the plane of and such that from a right-handed system.   Thus, is a unit vector to the plane of and . Note that is also a unit vector to the plane of and . Vectors of magnitude normal to the plane of and are given by . Example: 34 If is any vector, then is equal to (a) (b) 0 (c) (d) Solution: (d) Let = = = = Similarly and = . Example: 35 is equal to (a) (b) (c) (d) 1 Solution: (b) = = = = . Example: 36 The unit vector perpendicular to the vectors and , is (a) (b) (c) (d) Solution: (c) Let and = ; , which is a unit vector perpendicular to and . Example: 37 The sine of the angle between the vectors is (a) (b) (c) (d) Solution: (a) ; Example: 38 The vectors , and are such that a, c, b form a right handed system, then c is (a) (b) (c) (d) Solution: (a) form a right handed system. Hence,  = (7) Area of parallelogram and Triangle (i) The area of a parallelogram with adjacent sides and is . (ii) The area of a parallelogram with diagonals and is . (iii) The area of a plane quadrilateral ABCD is , where AC and BD are its diagonals. (iv) The area of a triangle with adjacent sides and is (v) The area of a triangle ABC is or or (vi) If are position vectors of vertices of a then its area = Note :  Three points with position vectors are collinear if Example: 39 The area of a triangle whose vertices are , and is (a) 13 (b) (c) 6 (d) Solution: (b) , Area of triangle ABC = = = Example: 40 If and , then the area of the parallelogram having diagonals and is (a) (b) (c) (d) Solution: (a) Area of the parallelogram with diagonals and = = = = = = Example: 41 The position vectors of the vertices of a quadrilateral ABCD are a, b, c and d respectively. Area of the quadrilateral formed by joining the middle points of its sides is (a) (b) (c) (d) Solution: (c) Let P, Q, R, S be the middle points of the sides of the quadrilateral ABCD. Position vector of P = , that of , that of R = and that of S = Mid point of diagonal Similarly mid point of PR As the diagonals bisect each other, PQRS is a parallelogram. ; Area of parallelogram PQRS = = = . 6.11 Moment of a Force and Couple. (1) Moment of a force (i) About a point : Let a force be applied at a point P of a rigid body. Then the moment of about a point O measures the tendency of to turn the body about point O. If this tendency of rotation about O is in anticlockwise direction, the moment is positive, otherwise it is negative. Let be the position vector of P relative to O. Then the moment or torque of about the point O is defined as the vector . If several forces are acting through the same point P, then the vector sum of the moment of the separate forces about O is equal to the moment of their resultant force about O. (ii) About a line: The moment of a force acting at a point P about a line L is a scalar given by where is a unit vector in the direction of the line, and , where O is any point on the line. Thus, the moment of a force about a line is the resolved part (component) along this line, of the moment of about any point on the line. Note :  The moment of a force about a point is a vector while the moment about a straight line is a scalar quantity. (2) Moment of a couple : A system consisting of a pair of equal unlike parallel forces is called a couple. The vector sum of two forces of a couple is always zero vector. The moment of a couple is a vector perpendicular to the plane of couple and its magnitude is the product of the magnitude of either force with the perpendicular distance between the lines of the forces. , where = , where is the angle between and = where is the arm of the couple and +ve or –ve sign is to be taken according as the forces indicate a counter-clockwise rotation or clockwise rotation.