VECTOR-02-THEORY-II

(3) Rotation about an axis : When a rigid body rotates about a fixed axis ON with an angular velocity , then the velocity of a particle P is given by , where and (unit vector along ON) Example: 42 Three forces and are acting on a particle at the point (0, 1, 2). The magnitude of the moment of the forces about the point (1, – 2, 0) is (a) (b) (c) (d) None of these Solution: (b) Total force Moment of the forces about P = Moment about P = = Magnitude of the moment = = Example: 43 The moment of the couple formed by the forces and acting at the points and respectively is (a) (b) (c) (d) Solution: (b) Moment of the couple, = = = 6.12 Scalar Triple Product . If are three vectors, then their scalar triple product is defined as the dot product of two vectors and . It is generally denoted by . or . It is read as box product of . Similarly other scalar triple products can be defined as . By the property of scalar product of two vectors we can say, (1) Geometrical interpretation of scalar triple product : The scalar triple product of three vectors is equal to the volume of the parallelopiped whose three coterminous edges are represented by the given vectors. form a right handed system of vectors. Therefore = volume of the parallelopiped, whose coterminous edges are a, b and c. (2) Properties of scalar triple product (i) If are cyclically permuted, the value of scalar triple product remains the same. i.e., or (ii) The change of cyclic order of vectors in scalar triple product changes the sign of the scalar triple product but not the magnitude i.e., (iii) In scalar triple product the positions of dot and cross can be interchanged provided that the cyclic order of the vectors remains same i.e., (iv) The scalar triple product of three vectors is zero if any two of them are equal. (v) For any three vectors and scalar , (vi) The scalar triple product of three vectors is zero if any two of them are parallel or collinear. (vii) If are four vectors, then (viii) The necessary and sufficient condition for three non-zero non-collinear vectors to be coplanar is that i.e., are coplanar . (ix) Four points with position vectors a, b, c and d will be coplanar, if . (3) Scalar triple product in terms of components (i) If , and be three vectors. Then, (ii) If and , then (iii) For any three vectors and (a) (b) (c) (4) Tetrahedron : A tetrahedron is a three-dimensional figure formed by four triangle OABC is a tetrahedron with as the base. and are known as edges of the tetrahedron. and are known as the pairs of opposite edges. A tetrahedron in which all edges are equal, is called a regular tetrahedron. Properties of tetrahedron (i) If two pairs of opposite edges of a tetrahedron are perpendicular, then the opposite edges of the third pair are also perpendicular to each other. (ii) In a tetrahedron, the sum of the squares of two opposite edges is the same for each pair. (iii) Any two opposite edges in a regular tetrahedron are perpendicular. Volume of a tetrahedron (i) The volume of a tetrahedron = = = = . Because are coplanar, so (ii) If are position vectors of vertices A, B and C with respect to O, then volume of tetrahedron OABC = (iii) If are position vectors of vertices A, B, C, D of a tetrahedron ABCD, then its volume = (5) Reciprocal system of vectors : Let be three non-coplanar vectors, and let . are said to form a reciprocal system of vectors for the vectors . If and form a reciprocal system of vectors, then (i) (ii) (iii) (v) are non-coplanar iff so are . Example: 44 If u, v and w are three non-coplanar vectors, then equals (a) 0 (b) (c) (d) Solution: (b) = = – = = . Example: 45 The value of ‘a’ so that the volume of parallelopiped formed by ; and becomes minimum is (a) – 3 (b) 3 (c) (d) Solution: (c) Volume of the parallelepiped V = = = = = ; ; At V is minimum at Example: 46 If be any three non-zero non-coplanar vectors, then any vector is equal to (a) (b) (c) (d) None of these Where , , Solution: (b) As are three non-coplanar vectors, we may assume =  But ; Similarly ; . Example: 47 If are non-coplanar vectors and is a real number, then the vectors and are non-coplanar for (a) No value of (b) All except one value of (c) All except two values of (d) All values of Solution: (c) As are non-coplanar vectors. Now, and will be non-coplanar iff i.e., i.e., Thus, given vectors will be non-coplanar for all values of except two values: and . Example: 48 x, y, z are distinct scalars such that where a, b, c are non-coplanar vectors then (a) (b) (c) (d) Solution: (a) are non-coplanar Now,     As ,    or But x, y, z are distinct. . 6.13 Vector Triple Product. Let be any three vectors, then the vectors and are called vector triple product of . Thus, (1) Properties of vector triple product (i) The vector triple product is a linear combination of those two vectors which are within brackets. (ii) The vector is perpendicular to and lies in the plane of and . (iii) The formula is true only when the vector outside the bracket is on the left most side. If it is not, we first shift on left by using the properties of cross product and then apply the same formula. Thus, = = (iv) If , and then Note :  Vector triple product is a vector quantity.  Example: 49 Let and be non-zero vectors such that . If is the acute angle between the vectors and , then equals (a) (b) (c) (d) Solution: (a) =    As and are not parallel, and   Example: 50 If and , then (a) 0 (b) 1 (c) 2 (d) 3 Solution: (a)   , = = = . Example: 51 If and are reciprocal system of vectors, then equals (a) (b) (c) (d) Solution: (c) , , Similarly and = = 6.14 Scalar product of Four Vectors. is a scalar product of four vectors. It is the dot product of the vectors and . It is a scalar triple product of the vectors and as well as scalar triple product of the vectors and . 6.15 Vector product of Four Vectors. (1) is a vector product of four vectors. It is the cross product of the vectors and . (2) are also different vector products of four vectors and . Example: 52 is equal to (a) (b) (c) (d) Solution: (d) = = = Example: 53 is equal to (a) (b) (c) (d) Solution: (c) = = = = Example: 54 Let the vectors a, b, c and d be such that . Let and be planes determined by pair of vectors and respectively. Then the angle between and is (a) (b) (c) (d) Solution: (a) 0  is parallel to Hence plane , determined by vectors is parallel to the plane determined by Angle between and = 0 (As the planes and are parallel). 6.16 Vector Equations. Solving a vector equation means determining an unknown vector or a number of vectors satisfying the given conditions. Generally, to solve a vector equation, we express the unknown vector as a linear combination of three known non-coplanar vectors and then we determine the coefficients from the given conditions. If are two known non-collinear vectors, then are three non-coplanar vectors. Thus, any vector where are unknown scalars. Example: 55 If and , then (a) (b) (c) (d) Solution: (a) Let Now,   Now,     . Thus Example: 56 The point of intersection of and where and is (a) (b) (c) (d) None of these Solution: (a) We have and Adding,   is parallel to = = For , Example: 57 Let . If is a unit vector such that , then is equal to (a) (b) (c) (d) Solution: (c) Let      .    ;    and . Example: 58 Let p, q, r be three mutually perpendicular vectors of the same magnitude. If a vector x satisfies equation , then x is given by (a) (b) (c) (d) Solution: (b) Let Let Now, = = = = – =     Example: 59 Let the unit vectors a and b be perpendicular and the unit vector c be inclined at an angle to both a and b. If , then (a) , (b) (c) (d) None of these Solution: (b) We have, ; (as ......(i) Taking dot product by ,   As ; Taking dot product of (i) by    Hence, Example: 60 The locus of a point equidistant from two given points whose position vectors are a and b is equal to (a) (b) (c) (d) Solution: (b) Let be a point on the locus.      . This is the locus of P.