VECTOR -03-ASSIGNMENT

1. The perimeter of a triangle with sides , and i(a) (b) (c) (d) 2. The magnitudes of mutually perpendicular forces a, b and c are 2, 10 and 11 respectively. Then the magnitude of its resultant is (a) 12 (b) 15 (c) 9 (d) None of these 3. If and , then the magnitude of (a) 13 (b) (c) (d) 4. The position vectors of A and B are 2i – 9j – 4k, and 6i – 3j + 8k respectively, then the magnitude of is (a) 11 (b) 12 (c) 13 (d) 14 5. If the position vectors of P and Q are and , then is (a) (b) (c) (d) 6. If a, b, c are mutually perpendicular unit vectors, then (a) (b) 3 (c) 1 (d) 0 7. Let and , then , holds for (a) All real p (b) No real p (c) p = –1 (d) p = 1 8. For any two vectors a and b, which of the following is true (a) (b) (c) (d) 9. If a and b are the adjacent sides of a parallelogram, then is a necessary and sufficient condition for the parallelogram to be a (a) Rhombus (b) Square (c) Rectangle (d) Trapezium 10. The direction cosines of vector in the direction of positive axis of x, is (a) (b) (c) (d) 11. A force is a (a) Unit vector (b) Localised vector (c) Zero vector (d) Free vector 12. A zero vector has (a) Any direction (b) No direction (c) Many directions (d) None of these 13. The perimeter of the triangle whose vertices have the position vectors , and is given by (a) (b) (c) (d) 14. If the vectors , and form a triangle, then it is (a) Right angled (b) Obtuse angled (c) Equilateral (d) Isosceles 15. The vectors and are the sides of a triangle ABC. The length of the median through A is (a) (b) (c) (d) 16. If a and b are two unit vectors inclined at an angle to each other, then , if (a) (b) (c) (d) 17. If the position vectors of A and B are and , then the direction cosine of along y- axis is (a) (b) (c) –5 (d) 11 18. The position vectors of four points A, B, C, D lying in plane are a, b, c, d respectively. They satisfy the relation , then the point D is (a) Centroid of (b) Circumcentre of (c) Orthocentre of (d) Incentre of 19. In a parallelopiped the ratio of the sum of the squares on the four diagonals to the sum of the squares on the three coterminous edges is (a) 2 (b) 3 (c) 4 (d) 1 20. P is the point of intersection of the diagonals of the parallelogram ABCD. If O is any point, then (a) (b) (c) (d) 21. If and , then the magnitude of p – 2q is (a) (b) 4 (c) (d) 22. If C is the middle point of AB and P is any point outside AB, then (a) (b) (c) (d) 23. If and , then the unit vector along a + b will be (a) (b) (c) (d) 24. What should be added in vector to get its resultant a unit vector i (a) (b) (c) (d) None of these 25. If , and , then the unit vector along its resultant is (a) (b) (c) (d) None of these 26. In the triangle ABC, , , then (a) (b) (c) (d) 27. If a has magnitude 5 and points north-east and vector b has magnitude 5 and points north-west, then (a) 25 (b) 5 (c) (d) 28. If , and , then (a) 6 (b) 5 (c) 4 (d) 3 29. If the sum of two unit vectors is a unit vector, then the angle between them is equal to (a) (b) (c) (d) 30. A, B, C, D, E are five coplanar points, then is equal to (a) (b) (c) (d) 31. If and , then the vectors a and b are (a) Parallel to each other (b) Perpendicular to each other (c) Inclined at an angle of 60o (d) Neither perpendicular nor parallel 32. If ABCDEF is a regular hexagon and ,then (a) 2 (b) 3 (c) 4 (d) 6 33. If O be the circumcentre and O' be the orthocentre of a triangle ABC, then (a) (b) (c) (d) 34. Let be a vector which makes an angle of 120o with a unit vector b. Then the unit vector (a + b) is (a) (b) (c) (d) 35. If be the angle between the unit vectors a and b, then (a) (b) (c) (d) 36. If , and , then the angle between a and b is (a) 0 (b) (c) (d) 37. If ABCD is a parallelogram, and , then the unit vector in the direction of BD is (a) (b) (c) (d) 38. If a and b are unit vectors making an angle with each other then is (a) 1 (b) 0 (c) (d) 39. If the moduli of the vectors a, b, c are 3, 4, 5 respectively and a and b + c, b and c + a, c and a + b are mutually perpendicular, then the modulus of a + b + c is (a) (b) 12 (c) (d) 50 40. If a and b are unit vectors and a – b is also a unit vector, then the angle between a and b is (a) (b) (c) (d) 41. If in a triangle , and D, E are the mid-points of AB and AC respectively, then is equal to (a) (b) (c) (d) 42. ABCDE is a pentagon. Forces act at a point. Which force should be added to this system to make the resultant (a) (b) (c) (d) 43. In a regular hexagon ABCDEF, (a) (b) (c) (d) None of these 44. (a) (b) (c) (d) None of these 45. In a triangle ABC, if , then equals (a) (b) (c) (d) None of these 46. If , then A, B, C form (a) Equilateral triangle (b) Right angled triangle (c) Isosceles triangle (d) Line 47. Three forces of magnitudes 1, 2, 3 dynes meet in a point and act along diagonals of three adjacent faces of a cube. The resultant force is (a) 114 dynes (b) 6 dynes (c) 5 dynes (d) None of these 48. If . Then angle between p and q is (a) (b) (c) (d) 49. If A, B, C are the vertices of a triangle whose position vectors are a, b, c and G is the centroid of the , then is (a) 0 (b) (c) (d) 50. If , and , then is (a) (b) (c) (d) 51. If x and y are two unit vectors and is the angle between them, then is equal to (a) 0 (b) (c) 1 (d) 52. If D, E, F are respectively the mid points of AB, AC and BC in , then (a) (b) (c) (d) 53. If ABCD is a rhombus whose diagonals cut at the origin O, then equals (a) (b) (c) (d) 54. ABCD is a parallelogram with AC and BD as diagonals. Then (a) (b) (c) (d) 55. The vectors b and c are in the direction of north-east and north-west respectively and . The magnitude and direction of the vector , are (a) , towards north (b) , towards west (c) 4, towards east (d) 4, towards south 56. Let a and b be two unit vectors inclined at an angle , then is equal to (a) (b) (c) (d) 57. If a, b, c are three vectors such that and the angle between b and c is , then (a) (b) (c) (d) (Note : Here ) 58. If are three vectors of equal magnitude and the angle between each pair of vectors is such that then is equal to (a) 2 (b) –1 (c) 1 (d) 59. Let be three unit vectors such that and . If c makes angles with a, b respectively then is equal to (a) (b) 1 (c) –1 (d) None of these 60. A vector of magnitude 2 along a bisector of the angle between the two vectors and is (a) (b) (c) (d) None of these 61. The vector is rotated through an angle and doubled in magnitude, then it becomes . The value of x is (a) (b) (c) (d) 2 62. If I is the centre of a circle inscribed in a triangle ABC, then is (a) 0 (b) (c) (d) None of these 63. If the vector bisects the angle between the vector and the vector then the unit vector in the direction of e is (a) (b) (c) (d) 64. The sides of a parallelogram are ,then the unit vector parallel to one of the diagonals (a) (b) (c) (d) 65. A point O is the centre of a cricle circumscribed about a triangle ABC. Then is equal to (a) (b) , where G is the centroid of triangle ABC (c) (d) None of these 66. If and a, b, c are non-coplanar, then the sum of (a) 0 (b) (c) (d) 67. Let a and b be two non-parallel unit vectors in a plane. If the vectors bisects the internal angle between a and b, then is (a) (b) 1 (c) 2 (d) 4 68. The horizontal force and the force inclined at an angle 60o with the vertical, whose resultant is in vertical direction of P kg, are (a) P, 2P (b) (c) (d) None of these 69. If the resultant of two forces is of magnitude P and equal to one of them and perpendicular to it, then the other force is (a) (b) P (c) (d) None of these 70. ABC is an isosceles triangle right angled at A. Forces of magnitude and 6 act along and respectively. The magnitude of their resultant force is (a) 4 (b) 5 (c) (d) 30 71. If the resultant of two forces of magnitudes P and Q acting at a point at an angle of 60o is , then is (a) 1 (b) 3/2 (c) 2 (d) 4 72. Five points given by A, B, C, D, E are in a plane. Three forces and act at A and three forces act at B. Then their resultant is (a) (b) (c) (d) 73. If a, b, c are the position vectors of the vertices A, B, C of the triangle ABC, then the centroid of is (a) (b) (c) (d) 74. If in the given figure , and , then (a) (b) (c) (d) 75. The position vectors of A and B are and . The position vector of the middle point of the line AB is (a) (b) (c) (d) None of these 76. If the position vectors of the points A and B are and , then what will be the position vector of the mid point of AB (a) (b) (c) (d) 77. The position vectors of two points A and B are and respectively. Then (a) 2 (b) 3 (c) 4 (d) 5 78. The position vector of the points which divides internally in the ratio 2 : 3 the join of the points and , is (a) (b) (c) (d) None of these 79. If a and b are P.V. of two points A, B, and C divides AB in ratio 2 : 1, then P.V. of C is (a) (b) (c) (d) 80. If three points A, B, C whose position vector are respectively and are collinear, then the ratio in which B divides AC is (a) 1 : 2 (b) 2 : 3 (c) 2 : 1 (d) 1 : 1 81. If O is the origin and C is the mid point of A (2, –1) and B (–4, 3). Then value of is (a) (b) (c) (d) 82. If the position vectors of P and Q are and respectively, then is equal to (a) (b) (c) (d) None of these 83. The position vectors of two vertices and the centroid of a triangle are and k respectively. The position vector of the third vertex of the triangle is (a) (b) (c) (d) None of these 84. The position vector of three consecutive vertices of a parallelogram are , and respectively. The position vector of the fourth vertex is (a) (b) (c) (d) None of these 85. If a and b are the position vectors of A and B respectively, then the position vector of a point C on AB produced such that is (a) 3a – b (b) 3b – a (c) 3a – 2b (d) 3b – 2a 86. If the position vectors of the points A, B, C, D be , , and respectively, then (a) (b) (c) (d) None of these 87. The position vector of a point C with respect to B is and that of B with respect to A is . The position vector of C with respect to A is (a) 2i (b) 2j (c) –2j (d) –2i 88. A and B are two points. The position vector of A is 6b – 2a. A point P divides the line AB in the ratio 1 : 2. If a – b is the position vector of P , then the position vector of B is given by (a) (b) (c) (d) 89. The points D, E, F divide BC, CA and AB of the triangle ABC in the ratio 1 : 4, 3 : 2 and 3 : 7 respectively and the point K divides AB in the ratio 1 : 3, then is equal to (a) 1 : 1 (b) 2 : 5 (c) 5 : 2 (d) None of these 90. The point B divides the arc AC of a quadrant of a circle in the ratio 1 : 2. If O is the centre and and , then the vector is (a) b – 2a (b) 2a – b (c) 3b – 2a (d) None of these 91. The point having position vectors are the vectices of (a) Right angled triangle (b) Isosceles triangle (c) Equilateral triangle (d) Collinear 92. Let p and q be the position vectors of P and Q respectively with respect to O and , The points R and S divide PQ internally and externally in the ratio 2 : 3 respectively. If and are perpendicular,then (a) (b) (c) (d) 93. The position vectors of the points A, B, C are and respectively. These points (a) Form an isosceles triangle (b) Form a right-angled triangle (c) Are collinear (d) Form a scalene triangle 94. ABCDEF is a regular hexagon where centre O is the origin. If the position vectors of A and B are and respectively, then is equal to (a) (b) (c) (d) None of these 95. Let and . If the point P on the line segment BC is equidistant from AB and AC, then is (a) (b) (c) (d) None of these 96. If and are the position vectors of the vertices A, B and C respectively of triangle ABC. The position vector of the point where the bisector of angle A meets BC, is (a) (b) (c) (d) 97. If and are parallel, then is (a) 4 (b) 2 (c) –2 (d) – 4 98. The vectors and are collinear, if (a) (b) (c) (d) 99. If a = (1, –1) and b = (–2, m) are two collinear vectors, then m = (a) 4 (b) 3 (c) 2 (d) 0 100. If a, b, c are the position vectors of three collinear points, then the existence of x, y, z is such that (a) (b) (c) (d) 101. If a and b are two non-collinear vectors, then = 0 (a) x = 0, but y is not necessarily zero (b) y = 0, but x is not necessarily zero (c) x = 0, y = 0 (d) None of these 102. If a and b are two non-collinear vectors, then (where x and y are scalars) represents a vector which is (a) Parallel to b (b) Parallel to a (c) Coplanar with a and b (d) None of these 103. If a, b, c are non-collinear vectors such that for some scalars x, y, z, , then (a) (b) (c) (d) 104. If the position vectors of the points A, B, C be a, b, 3a – 2b respectively, then the points A, B, C are (a) Collinear (b) Non-collinear (c) Form a right angled triangle (d) None of these 105. If two vertices of a triangle are and , then the third vertex can be (a) (b) (c) (d) 106. If the vectors and are parallel, then the value of x and y will be (a) –1, –2 (b) 1, –2 (c) –1, 2 (d) 1, 2 107. The position vectors of four points P, Q, R, S are and respectively, then (a) is parallel to (b) is not parallel to (c) is equal to (d) is parallel and equal to 108. The vectors and have their initial points at (1, 1). The value of so that the vectors terminate on one straight line, is (a) 0 (b) 3 (c) 6 (d) 9 109. The points with position vectors and are collinear. The value of p is (a) 7 (b) – 37 (c) – 7 (d) 37 110. Three points whose position vectors are and will be collinear, if the value of k is (a) Zero (b) Only negative real number (c) Only positive real number (d) Every real number 111. The points with position vectors and are collinear, If a = (a) – 8 (b) 4 (c) 8 (d) 12 112. Let the value of and , where a and b are non-collinear vectors. If , then the value of x and y will be (a) – 1, 2 (b) 2, –1 (c) 1, 2 (d) 2, 1 113. If and , then the value of will be (a) – 2, 0 (b) 0, – 2 (c) – 1, 0 (d) 0, – 1 114. The vectors are collinear, if equals (a) 3 (b) 4 (c) 5 (d) 6 115. If three points A, B and C have position vectors (1, x, 3), (3, 4, 7) and (y, – 2, – 5) respectively and if they are collinear, then (x, y) = (a) (2, – 3) (b) (– 2, 3) (c) (2, 3) (d) (– 2, – 3) 116. The position vectors of three points are and where a, b, c are non-coplanar vectors. The points are collinear when (a) (b) (c) (d) None of these 117. Three points whose position vectors are a, b, c will be collinear if (a) (b) (c) (d) None of these 118. If and , then a vector along r which is linear combination of p and q and also perpendicular to q is (a) (b) (c) (d) None of these 119. If a and b are two non zero and non-collinear vectors, then a + b and a – b are (a) Linearly dependent vectors (b) Linearly independent vectors (c) Linearly dependent and independent vectors (d) None of these 120. If p, q are two non-collinear and non-zero vectors such that , where a, b, c are the lengths of the sides of a triangle, then the triangle is (a) Right angled (b) Obtuse angled (c) Equilateral (d) Isosceles 121. If and such that then (a) are in A.P. (b) are in A.P. (c) are in H.P. (d) are in G.P. 122. Let a, b, c are three non-coplanar vectors such that . If , then (a) (b) (c) (d) 123. If and then c and a are (a) Like parallel vectors (b) Unlike parallel vectors (c) Are at right angles (d) None of these 124. The sides of a triangle are in A.P., then the line joining the centroid to the incentre is parallel to (a) The largest side (b) The smaller side (c) The middle side (d) None of the sides 125. In a trapezoid the vector . We will then find that is collinear with AD. If , then (a) (b) (c) (d) 126. The angle between the vectors and is (a) 0 (b) (c) (d) 127. If and , then is perpendicular to c if t = (a) 2 (b) 4 (c) 6 (d) 8 128. The angle between the vectors and is (a) (b) (c) (d) 129. If a, b, c are non zero-vectors such that a . b = a . c, then which statement is true (a) (b) (c) or (d) None of these 130. The vector is perpendicular to the vector , if a = (a) 5 (b) – 5 (c) – 3 (d) 3 131. The vector is perpendicular to the vector , if (a) 0 (b) – 1 (c) – 2 (d) – 3 132. If the vectors and are perpendicular to each other, then a is given by (a) 9 (b) 16 (c) 25 (d) 36 133. The value of for which the vectors and are perpendicular, is (a) None (b) – 1 (c) 1 (d) Any 134. The angle between the vectors and is (a) (b) (c) (d) 135. If is a unit vector perpendicular to plane of vector a and b and angle between them is , then a . b will be (a) (b) (c) (d) 136. If the vectors and are perpendicular, then (a) (b) (c) (d) 137. If be the angle between two vectors a and b, then if (a) (b) (c) (d) None of these 138. If and and , then value of will be (a) 2 (b) – 1 (c) – 2 (d) 1 139. If a and b are mutually perpendicular vectors, then (a) (b) (c) (d) 140. If and , then the angle between the vectors and is (a) 30o (b) 60o (c) 90o (d) 0o 141. , then (a) (b) (c) Angle between a and b is 60o (d) None of these 142. The angle between the vectors and is (a) (b) (c) (d) 143. If the vectors and are perpendicular to each other, then a = (a) 6 (b) – 6 (c) 5 (d) – 5 144. If the angle between two vectors and is , then the value of a = (a) 2 (b) 4 (c) –2 (d) 0 145. (a . b) c and (a . c) b are (a) Two like vectors (b) Two equal vectors (c) Two vectors in direction of a (d) None of these 146. The angle between the vector and is (a) (b) (c) (d) 0 147. If a = (1, –1, 2), b = (– 2, 3, 5 ), c = (2, – 2, 4) and i is the unit vector in the x-direction, then (a) 11 (b) 15 (c) 18 (d) 36 148. If and are perpendicular vectors, then the value of a is (a) 5 (b) – 5 (c) 7 (d) 149. If and then (a) (b) (c) (d) 150. If a and b are adjacent sides of a rhombus, then (a) a . b = 0 (b) (c) (d) None of these 151. If , then is (a) Positive (b) Negative (c) Zero (d) None of these 152. If and are at right angle, then m = (a) – 6 (b) – 8 (c) – 10 (d) – 12 153. If the vectors and are perpendicular, then is (a) – 14 (b) 7 (c) 14 (d) 154. is equal to (a) (b) 3 (c) (d) None of these 155. If the vectors and are orthogonal to each other, then the locus of the point (x, y) is (a) A circle (b) An ellipse (c) A parabola (d) A straight line 156. If , and , then the value of t such that is at right angle to vector , is (a) 3 (b) 4 (c) 5 (d) 6 157. If a and b are two perpendicular vectors, then out of the following four statements (i) (ii) (iii) (iv) (a) Only one is correct (b) Only two are correct (c) Only three are correct (d) All the four are correct 158. If a, b, c are unit vectors such that , then (a) 1 (b) 3 (c) (d) 159. A unit vector in the xy-plane which is perpendicular to is (a) (b) (c) (d) None of these 160. The vectors and are perpendicular, when (a) (b) (c) (d) None of these 161. The unit normal vector to the line joining and and pointing towards the origin is 162. The position vector of coplanar points A, B, C, D are a, b, c and d respectively, in such a way that , then the point D of the triangle ABC is (a) Incentre (b) Circumcentre (c) Orthocentre (d) None of these 163. If , and , then the scalar product of and will be (a) 3 (b) 6 (c) 9 (d) 12 164. If the moduli of a and b are equal and angle between them is 120o and a . b = – 8, then is equal to (a) – 5 (b) – 4 (c) 4 (d) 5 165. The position vector of vertices of a triangle ABC are and respectively, then (a) (b) (c) (d) 166. A, B, C, D are any four points, then (a) (b) (c) (d) 0 167. If and , then (a) – 13 (b) – 10 (c) 13 (d) 10 168. The value of c so that for all real x, the vectors make an obtuse angle are (a) c < 0 (b) (c) (d) 169. The vector is (a) A unit vector (b) Makes an angle with the vector (c) Parallel to the vector (d) Perpendicular to the vector 170. The value of x for which the angle between the vectors and is acute and the angle between b and x-axis lies between and satisfy (a) (b) (c) only (d) only 171. If the scalar product of the vector with a unit vector parallel to the sum of the vectors and be 1, then (a) 1 (b) – 1 (c) 2 (d) – 2 172. If a is any vector in space, then (a) (b) (c) (d) 173. If a, b and c are unit vectors, then does not exceed (a) 4 (b) 9 (c) 8 (d) 6 174. If a and b are two unit vectors, such that and are perpendicular to each other then the angle between a and b is (a) 45o (b) 60o (c) (d) 175. a, b, c are three vectors, such that , , then is equal to (a) 0 (b) – 7 (c) 7 (d) 1 176. A unit vector in xy-plane that makes an angle 45o with the vectors and an angle of 60o with the vector is (a) i (b) (c) (d) None of these 177. The angle between the vectors and , when a = (1, 1, 4) and b = (1, – 1, 4) is (a) 90o (b) 45o (c) 30o (d) 15o 178. Let and . If n is a unit vector such that u . n = 0 and v . n = 0, then is equal to (a) 0 (b) 1 (c) 2 (d) 3 179. If a, b, c are the pth, qth, rth terms of an HP and , then (a) u, v are parallel vectors (b) u, v are orthogonal vectors (c) u . v = 1 (d) 180. ABC is an equilateral triangle of side a. The value of is equal to (a) (b) (c) (d) None of these 181. If and and a and b are two vectors such that and then angle between a and b is (a) (b) (c) (d) 182. A vector whose modulus is and makes the same angle with , and , will be (a) (b) (c) (d) 183. In a right angled triangle ABC, the hypotenues AB = p, then is equal to (a) (b) (c) (d) None of these 184. If the vectors and are orthogonal and a vector makes an obtuse angle with the z-axis, then the value of is (a) (b) (c) (d) 185. If the vectors and are inclined at an acute angle, then (a) (b) (c) (d) None of these 186. The value of x for which the angle between the vectors and is acute and the angle between the vector b and y-axis lies between and are (a) < 0 (b) > 0 (c) – 2, – 3 (d) 1, 2 187. If a, b, c are linearly independent vectors and , then (a) (b) (c) any non-zero value (d) None of these 188. The position vectors of the points A, B and C are and respectively.The greatest angle of the triangle ABC is (a) (b) 90o (c) (d) 189. If a and b are two non-zero vectors, then the component of b along a is (a) (b) (c) (d) 190. Projection of the vector in the direction of the vector will be (a) (b) (c) (d) 191. If and , then the component of a along b is (a) (b) (c) (d) 192. The projection of vector on the vector will be (a) (b) (c) (d) 193. If vector and vector , then (a) (b) (c) 3 (d) 7 194. The projection of a along b is (a) (b) (c) (d) 195. If and , then the projection of b on a is (a) 3 (b) 4 (c) 5 (d) 6 196. The projection of the vector along the vector j is (a) 1 (b) 0 (c) 2 (d) – 1 197. If is a unit vector and b, a non-zero vector not parallel to , then the vector is (a) Parallel to b (b) At right angles to (c) Parallel to (d) At right angles to b 198. If and , then a vector in the direction of a and having magnitude as is (a) (b) (c) (d) None of these 199. The vector is to be written as the sum of a vector parallel to and a vector perpendicular to a. Then (a) (b) (c) (d) 200. The components of a vector a along and perpendicular to the non-zero vector b are respectively (a) (b) (c) (d) 201. Let and c be two vectors perpendicular to each other in the xy-plane. All vectors in the same plane having projections 1 and 2 along b and c respectively, are given by (a) (b) (c) (d) 202. Let and be three vectors. A vector in the plane of b and c whose projection on a is of magnitude is (a) (b) (c) (d) 203. If the position vectors of A and B be 6 and , then the work done by the force in displacing a particle from A to B is (a) 15 units (b) 17 units (c) – 15 units (d) None of these 204. If the force moves from to , then work done will be represented by (a) 3 (b) 4 (c) 5 (d) 6 205. The work done by the force in displacing a particle from the point (3, 4, 5) to the point (1, 2, 3) is (a) 2 units (b) 3 units (c) 4 units (d) 5 units 206. The work done in moving an object along the vector , if the applied force is , is (a) 7 (b) 8 (c) 9 (d) 10 207. A force acts at a point A whose position vector is . If point of application of moves from A to the point B with position vector , then work done by is (a) 4 (b) 20 (c) 2 (d) None of these 208. Force and are acting on a particle and displace it from the point to the point , then work done by the force is (a) 30 units (b) 36 units (c) 24 units (d) 18 units 209. A force of magnitude 5 units acting along the vector displaces the point of application from (1, 2, 3) to (5, 3, 7), then the work done is (a) 50/7 (b) 50/3 (c) 25/3 (d) 25/4 210. If forces of magnitudes 6 and 7 units acting in the directions and respectively act on a particle which is displaced from the point P(2, –1, –3) to Q(5, –1, 1), then the work done by the forces is (a) 4 units (b) – 4 units (c) 7 units (d) – 7 units 211. If and , then a unit vector perpendicular to both u and v is (a) (b) (c) (d) None of these 212. (a) (b) (c) (d) None of these 213. If , then which relation is correct (a) (b) (c) (d) None of these 214. If be the angle between the vectors a and b and , then (a) (b) (c) (d) 0 215. If a and b are two vectors such that a . b = 0 and ,then (a) a is parallel to b (b) a is perpendicular to b (c) Either a or b is a null vector (d) None of these 216. (a) (b) (c) (d) 217. Which of the following is not a property of vectors (a) (b) (c) (d) 218. The number of vectors of unit length perpendicular to vectors a = (1, 1, 0) and b = (0, 1, 1) is (a) Three (b) One (c) Two (d) Infinite 219. If , then true statement is (a) (b) (c) (d) 220. A unit vector which is perpendicular to and to is (a) (b) (c) (d) 221. The unit vector perpendicular to the and , is (a) (b) (c) (d) 222. The sine of the angle between the two vectors and will be (a) (b) (c) (d) None of these 223. For any two vectors a and b, if , then (a) a = 0 (b) b = 0 (c) Not parallel (d) None of these 224. For any vectors a, b, c. (a) 0 (b) (c) [a b c] (d) 225. If and , then a . b is equal to (a) 0 (b) 2 (c) 4 (d) 6 226. If and ,then the value of is (a) (b) (c) (d) 227. A unit vector perpendicular to the vector and is (a) (b) (c) (d) 228. A unit vector perpendicular to each of the vector and is equal to (a) (b) (c) (d) 229. If and , then the unit vector perpendicular to a and b is (a) (b) (c) (d) 230. If is the angle between the vectors a and b, then equal to (a) (b) (c) (d) 231. A vector perpendicular to both of the vectors and is (a) (b) (c) is a scalar (d) None of these 232. A unit vector perpendicular to the plane of is (a) (b) (c) (d) 233. The unit vector perpendicular to the both the vectors and is (a) (b) (c) (d) None of these 234. The unit vector perpendicular to the vectors and is (a) (b) (c) (d) 235. If and , then is (a) (b) (c) (d) 236. If and the angle between a and b is , then is equal to (a) 48 (b) 16 (c) a (d) None of these 237. and are two vectors and c is a vector such that , then is (a) (b) (c) 34 : 39 : 45 (d) 39 : 35 : 34 238. , then (a) (b) (c) (d) 239. If and , then is (a) (b) (c) (d) 240. The unit vector perpendicular to both i + j and j + k is (a) (b) (c) (d) 241. If and , then (a) 2 (b) 6 (c) 8 (d) 20 242. If and , then (a) 16 (b) 8 (c) 3 (d) 12 243. The unit vector perpendicular to both the vectors and and making an acute angle with the vector k is (a) (b) (c) (d) None of these 244. The angle between and is (a) 30o (b) 60o (c) 90o (d) 245. If the vectors a, b and c are represented by, the sides BC, CA and AB respectively of the , then (a) (b) (c) (d) 246. , where a, b and c are coplanar vectors, then for some scalar k (a) (b) (c) (d) None of these 247. If and , then (a) (b) (c) (d) None of these 248. If a and b are two vectors, then equals (a) (b) (c) (d) None of these 249. Given and . A unit vector perpendicular to both and is (a) i (b) j (c) k (d) 250. For any two vectors a and b, is equal to (a) (b) (c) (d) None of these 251. If vectors and form a left handed system, then is (a) (b) (c) (d) 252. is equal to (a) 3r (b) r (c) 0 (d) None of these 253. If a, b, c are noncoplanar vectors such that and , then (a) (b) (c) (d) None of these 254. If and ,then the length of the perpendicular from A to the line BC is (a) (b) (c) (d) None of these 255. The area of a parallelogram whose two adjacent sides are represented by the vector and is (a) (b) (c) (d) 256. The area of the parallelogram whose diagonals are and is (a) (b) (c) 8 (d) 4 257. The area of a parallelogram whose diagonals coincide with the following pair of vectors is . The vectors are (a) (b) (c) (d) None of these 258. If and represents the adjacent sides of a parallelogram, then the area of this parallelogram is (a) (b) (c) (d) 259. If the vectors represents the diagonals of a parallelogram, then its area will be (a) (b) (c) (d) 260. The area of a parallelogram whose adjacent sides are and , is (a) (b) (c) (d) 261. If the diagonals of a parallelogram are represented by the vectors and , then its area in square units is (a) (b) (c) (d) 262. The area of a parallelogram whose adjacent sides are given by the vectors and (in square units) is (a) (b) (c) (d) 263. The area of the parallelogram whose diagonals are and is (a) (b) (c) (d) 264. The area of the triangle whose two sides are given by and is (a) 17 (b) (c) (d) 265. If and are the vector sides of any triangle, then its area is given by (a) 41 (b) 47 (c) (d) 266. Let a, b, c be the position vectors of the vertices of a triangle ABC. The vector area of triangle ABC is (a) (b) (c) (d) 267. Consider a tetrahedron with faces . Let be the vectors whose magnitudes are respectively equal to areas of and whose directions are perpendicular to these faces in outward direction. Then equals (a) 1 (b) 4 (c) 0 (d) None of these 268. A unit vector perpendicular to the plane determined by the points (1, –1, 2), (2, 0, –1) and (0, 2, 1) is (a) (b) (c) (d) 269. The position vectors of the points A, B and C are , and respectively. The vector area of the , where (a) (b) (c) (d) 270. The area of the triangle having vertices as is (a) 26 (b) 11 (c) 36 (d) 0 271. Let and , where O, A and C are noncollinear points. Let p denote the area of the quadrilateral OABC, and q denote the area of the parallelogram with OA and OC as adjacent sides. Then is equal to (a) 4 (b) 6 (c) (d) None of these 272. The adjacent sides of a parallelogram are along and . The angles between the diagonals are (a) 30o and 150o (b) 45o and 135o (c) 90o and 90o (d) None of these 273. Four points with position vectors and form a (a) Rhombus (b) Parallelogram but not rhombus (c) Rectangle (d) Square 274. In a . If the area of triangle is of unit magnitude, then (a) (b) (c) (d) 275. The moment of the force acting at a point P, about the point C is (a) (b) (c) A vectors having the same direction as (d) 276. A force acts at a point A, whose position vector is . The moment of about the origin is (a) (b) (c) (d) 277. Let the point A, B, and P be (–2, 2, 4), (2, 6, 3) and (1, 2, 1) respectively. The magnitude of the moment of the force represented by and acting at A about P is (a) 15 (b) (c) (d) None of these 278. The moment about the point M(–2, 4, –6) of the force represented in magnitude and position by where the points A and B have the coordinates (1, 2, –3) and (3, –4, 2) respectively, is (a) (b) (c) (d) 279. A force of 39 kg. wt is acting at a point P (–4, 2, 5) in the direction of . The moment of this force about a line through the origin having the direction of is (a) 76 units (b) –76 units (c) (d) None of these 280. If the magnitude of moment about the point of a force acting through the point is , then the value of is (a) 1 (b) 2 (c) 3 (d) 4 281. is equal to (a) (b) (c) (d) 0 282. If a, b, c are three non-coplanar vector, then (a) 0 (b) 2 (c) –2 (d) None of these 283. If i, j, k are the unit vectors and mutually perpendicular, then is equal to (a) 0 (b) –1 (c) 1 (d) None of these 284. If and , then (a) 6 (b) 10 (c) 12 (d) 24 285. If or a, b, c are a right handed triad of mutually perpendicular vectors, then [a b c] = (a) (b) 1 (c) –1 (d) A non-zero vector 286. (a) 1 (b) 3 (c) –3 (d) 0 287. If , then = (a) 3a (b) (c) 0 (d) None of these 288. If , then (a) 12 (b) 2 (c) 0 (d) – 12 289. (a) b . b (b) (c) 0 (d) 290. For three vectors u, v, w which of the following expressions is not equal to any of the remaining three (a) (b) (c) (d) 291. Which of the following expressions are meaningful (a) (b) (c) (d) 292. Given vectors a, b, c such that , the value of is (a) 3 (b) 1 (c) (d) 293. If and , then is (a) 122 (b) – 144 (c) 120 (d) – 120 294. is equal to (a) (b) (c) (d) None of these 295. (a) 1 (b) 3 (c) – 3 (d) – 1 296. If , then is equal to (a) 3 (b) 1 (c) – 1 (d) None of these 297. If the vectors and form three concurrent edges of a parallelopiped, then the volume of the parallelopiped is (a) 8 (b) 10 (c) 4 (d) 14 298. If three vectors and represents a cube, then its volume will be (a) 616 (b) 308 (c) 154 (d) None of these 299. Volume of the parallelopiped whose coterminous edges are , is (a) 5 cubic units (b) 6 cubic units (c) 7 cubic units (d) 8 cubic units 300. If and are the three coterminous edges of a parallelopiped, then its volume is (a) 108 (b) 210 (c) 272 (d) 308 301. Three concurrent edges OA, OB, OC of a parallelopiped are represented by three vectors and , the volume of the solid so formed in cubic units is (a) 5 (b) 6 (c) 7 (d) 8 302. What will be the volume of that parallelopiped whose sides are and (a) 5 unit (b) 6 unit (c) 7 unit (d) 8 unit 303. The volume of the parallelopiped whose coterminous edges are and is (a) 4 (b) 3 (c) 2 (d) 8 304. The volume of the parallelopiped whose edges are represented by and is 546, then (a) 3 (b) 2 (c) – 3 (d) – 2 305. , if (a) (b) (c) (d) 306. If a, b, c be any three non-coplanar vectors, then (a) [a b c] (b) 2[a b c] (c) [a b c]2 (d) 2[a b c]2 307. If a, b, c are three non-coplanar vectors and p, q, r are defined by the relations , , , then = (a) 0 (b) 1 (c) 2 (d) 3 308. If , , , where a, b, c are three non-coplanar vectors, then the value of is given by (a) 3 (b) 2 (c) 1 (d) 0 309. The value of , where and is (a) 0 (b) 1 (c) 2 (d) 4 310. If a, b and c are three non-coplanar vectors, then is equal to (a) [a b c] (b) 2[a b c] (c) – [a b c] (d) 0 311. If a, b, c are three coplanar vectors, then (a) [a b c] (b) 2[a b c] (c) 3[a b c] (d) 0 312. If b and c are any two non-collinear unit vectors and a is any vector, then (a) a (b) b (c) c (d) 0 313. If three coterminous edges of a parallelopiped are represented by and , then its volume is (a) [a b c] (b) 2[a b c] (c) [a b c]2 (d) 0 314. If a, b and c are unit coplanar vectors then the scalar triple product is equal to (a) 0 (b) 1 (c) (d) 315. Let and , then depends on (a) Only x (b) Only y (c) Neither x nor y (d) Both x and y 316. (a) – [a b c] (b) [a b c] (c) 0 (d) 2[a b c] 317. Let and if U is a unit vector, then the maximum value of the scalar triple product [U V W] is (a) – 1 (b) (c) (d) 318. If a, b are non-zero and non-collinear vectors then is equal to (a) (b) (c) (d) 319. If a, b, c are three non-coplanar nonzero vectors then is equal to (a) [b c a]a (b) [c a b]b (c) [a b c]c (d) None of these 320. Let a, b, c be three unit vectors and . If the angle between b and c is , then is equal to (a) (b) (c) 1 (d) None of these 321. If a, b, c are three non-coplanar vectors represented by concurrent edges of a parallelopiped of volume 4, then is equal to (a) 12 (b) 4 (c) (d) 0 322. The three concurrent edges of a parallelopiped represent the vectors a, b, c such that . Then the volume of the parallelopiped whose three concurrent edges are the three concurrent diagonals of three faces of the given parallelopiped is (a) (b) (c) (d) None of these 323. If a, b, c are non-coplanar non-zero vectors and r is any vector in space then [b c r]a+[c a r]b+[a b r]c is equal to (a) 3[a b c]r (b) [a b c]r (c) [b c a]r (d) None of these 324. If the vertices of a tetrahedron have the position vectors 0, i + j, 2j – k and then the volume of the tetrahedron is (a) (b) 1 (c) 2 (d) None of these 325. The three vectors taken two at a time form three planes. The three unit vectors drawn perpendicular to three planes form a parallelopiped of volume (a) cubic units (b) 4 cubic units (c) cubic units (d) cubic units 326. The volume of the tetrahedron whose vertices are the points with position vectors and is 11 cubic units if the value of is (a) – 1 (b) 1 (c) – 7 (d) 7 327. Let a, b and c be three non-zero and non-coplanar vectors and p, q and r be three vectors given by and . If the volume of the parallelopiped determined by a, b, and c is V1 and that of the parallelopiped determined by p, q, and r is V2 , then V2 : V1 = (a) 2 : 3 (b) 5 : 7 (c) 15 : 1 (d) 1 : 1 328. If a, b, c are any three vectors and their inverse are and , then will be (a) Zero (b) One (c) Non-zero (d) [a b c] 329. a, b, c are three non-zero, non-coplanar vectors and p, q, r are three other vectors such that , , . Then [p q r] equals (a) (b) (c) 0 (d) None of these 330. If the vectors , and be coplanar, then (a) – 1 (b) – 2 (c) – 3 (d) – 4 331. If and are coplanar then the value of p will be (a) – 6 (b) – 2 (c) 2 (d) 6 332. A unit vector which is coplanar to vector and and perpendicular to , is (a) (b) (c) (d) 333. If the vectors and are coplanar, then x = (a) (b) (c) 0 (d) 1 334. If the vectors and are coplanar ,then the value of x is (a) – 2 (b) 2 (c) 1 (d) 3 335. are coplanar, then the value of is (a) (b) (c) (d) None of these 336. . If and , then to make three vectors coplanar (a) (b) (c) (d) No value of can be found 337. The vector a lies in the plane of vectors b and c, which of the following is correct (a) (b) (c) (d) 338. If the vectors ; are coplanar, then is equal to (a) 0 (b) 1 (c) 2 (d) Not defined 339. If and and are non-coplanar vectors, then abc is equal to (a) – 1 (b) 0 (c) 1 (d) 4 340. Let a, b, c be distinct non-negative numbers. If the vectors and lie in a plane, then c is (a) The airthmetic mean of a and b (b) The geometric mean of a and b (c) The harmonic mean of a and b (d) Equal to zero 341. If the vectors and are coplanar, then the value of (a) – 1 (b) (c) (d) 1 342. If a, b, c are position vector of vertices of a triangle ABC, then unit vector perpendicular to its plane is (a) (b) (c) (d) None of these 343. If a, b, c are non-coplanar vectors and , then is equal to (a) (b) (c) (d) 344. If the points whose position vectors are and lie on a plane, then (a) (b) (c) (d) 345. Vector coplanar with vectors i+j and j+k and parallel to the vector , is (a) (b) (c) (d) 346. Let and . Then (a) (b) are coplanar (c) (d) None of these 347. Let and . If c is parallel to the plane of the vectors a and b then is (a) 1 (b) 0 (c) – 1 (d) 2 348. The vectors and are coplanar for (a) All values of x (b) x < 0 (c) x > 0 (d) None of these 349. Given a cube ABCD with lower base ABCD, upper base and the lateral edges and ; M and are the centres of the faces ABCD and respectively. O is a point on line , such that , then , if (a) (b) (c) (d) 350. is equal to (a) 0 (b) i (c) j (d) k 351. If and , then is equal to (a) (b) (c) (d) None of these 352. If , then is (a) (b) (c) (d) 353. If a, b, c are any three vectors then is a vector (a) Perpendicular to (b) Coplanar with a and b (c) Parallel to c (d) Parallel to either a or b 354. If a and b are two unit vectors, then the vector is parallel to the vector (a) (b) (c) (d) 355. If , then (a) (b) (c) (d) 356. Which of the following is a true statement (a) is coplanar with c (b) is perpendicular to a (c) is perpendicular to b (d) is perpendicular to c 357. If , then (a) (b) (c) (d) 358. A unit vector perpendicular to vector c and coplanar with vectors a and b is (a) (b) (c) (d) None of these 359. equals (a) i (b) j (c) k (d) 0 360. Given three unit vectors a, b, c such that and , then is (a) a (b) b (c) c (d) 0 361. , then is (a) (b) (c) (d) 362. If i, j, k are unit vectors, then (a) (b) (c) (d) 363. If a, b, c are any vectors, then the true statement is (a) (b) (c) (d) 364. is equal to (a) (b) (c) (d) 365. (a) 0 (b) (c) (d) 366. If a, b, c are non-coplanar unit vectors such that , then the angle between a and b is (a) (b) (c) (d) 367. Let a, b, c be three vectors from , if (a) (b) (c) (d) 368. If a, b, c are any three vectors such that , then is (a) 0 (b) a (c) b (d) None of these 369. If three unit vectors a, b, c are such that , then the vector a makes with b and c respectively the angles (a) 40o, 80o (b) 45o, 45o (c) 30o, 60o (d) 90o, 60o 370. and are (a) Linearly dependent (b) Equal vectors (c) Parallel vectors (d) None of these 371. a and b are two given vectors. On these vectors as adjacent sides a parallelogram is constructed. The vector which is the altitude of the parallelogram and which is perpendicular to a is (a) (b) (c) (d) 372. If and , then is equal to (a) 60 (b) 64 (c) 74 (d) – 74 373. (a) [b c a]a (b) [c a b]b (c) [a b c ]c (d) [a c b]b 374. If a, b, c, d are coplanar vectors, then (a) (b) (c) (d) 0 375. If . and represent dot product and cross product respectively then which of the following is meaningless (a) (b) (c) (d) 376. Two planes are perpendicular to one another. One of them contains vectors a and b and the other contains vectors c and d, then equals (a) 1 (b) 0 (c) (d) [b c d] 377. a, b, c, d are any four vectors then is a vector (a) Perpendicular to a, b, c, d (b) Along the line of intersection of two planes, one containing a, b and the other containing c, d (c) Equally inclined to both and (d) None of these 378. If a, b, c are non-coplanar non-zero vectors then is equal to (a) (b) (c) 0 (d) None of these 379. If a, b, c are three non-coplanar non-zero vectors and r is any vector is space then is equal to (a) 2[a b c]r (b) 3[a b c]r (c) [a b c]r (d) None of these 380. If then is equal to (a) (b) (c) (d) None of these 381. is equal to (a) (b) (c) (d) 382. equals (a) (b) (c) (d) None of these 383. is equal to (a) [a b c]2 (b) [a b c]3 (c) [a b c]4 (d) None of these 384. If a, b, c are coplanar vectors, then (a) (b) (c) (d) 385. For any three non-zero vectors r1, r2 and r3, . Then which of the following is false (a) All the three vectors are parallel to one and the same plane (b) All the three vectors are linearly dependent (c) This system of equation has a non-trivial solution (d) All the three vectors are perpendicular to each other 386. is equal to (a) (b) (c) (d) None of these 387. If a, b, c are vectors such that [a b c] = 4, then (a) 16 (b) 64 (c) 4 (d) 8 388. If position vector of points A, B, C are respectively i, j, k and AB = CX, then position vector of point X is (a) (b) (c) (d) 389. If , then a = (a) i (b) k (c) j (d) 390. If and , then is equal to (a) (b) (c) (d) None of these 391. Given that the vectors a and b are non-collinear, the values of x and y for which the vector equality holds true if are (a) (b) (c) (d) 392. If and , then (a) (b) (c) (d) None of these 393. If where a, b, c are non-coplanar, then (a) (b) (c) (d) 394. If i, j, k are unit orthonormal vectors and a is a vector, if , then is (a) 0 (b) 1 (c) – 1 (d) Arbitrary scalar 395. where implies that (a) b = c (b) a and b are parallel (c) a, b, c are mutually perpendicular (d) a, b, c are coplanar 396. The scalars l and m such that , where a, b and c are given vectors, are equal to (a) (b) (c) (d) None of these 397. If a is a vector perpendicular to the vectors and and satisfies the condition , then a = (a) (b) (c) (d) None of these 398. If a = (1, – 1, 1) and c = (– 1, – 1, 0), then the vector b satisfying and is (a) (1, 0, 0) (b) (0, 0, 1) (c) (0, – 1, 0) (d) None of these 399. If a = (1, 1, 1), c = (0, 1, – 1) are two vectors and b is a vector such that and , then b is equal to (a) (b) (c) (5, 2, 2) (d) 400. If and . If and , then d will be (a) (b) (c) (d) 401. If and , then (a) b = 0 (b) (c) (d) None of these 402. If and for some non-zero vector x, then the true statement is (a) [a b c] = 0 (b) (c) [a b c] = 1 (d) None of these 403. A unit vector a makes an angle with z-axis. If is a unit vector, then a is equal to (a) (b) (c) (d) None of these 404. If , then (a) 8 (b) – 8 (c) 2 (d) 0 405. Given the following simultaneous equations for vectors x and y ............(i) ............(ii) ............(iii). Then x =............., y = ............. (a) a, a – x (b) a – b, b (c) b, a – b (d) None of these 406. a is not perpendicular to b, then r = (a) a – b (b) a +b (c) (d) 407. Let and . If c is a vector such that and the angle between and c is 30o , then (a) (b) (c) 2 (d) 3 408. Let and a unit vector c be coplanar. If c is perpendicular to a, then c = (a) (b) (c) (d) 409. Let a and b be two non-collinear unit vectors. If and , then is (a) (b) (c) (d) 410. Let a, b, c be three vectors such that and and . If , then equal to (a) 1 (b) – 4 (c) 4 (d) – 2 411. Unit vectors a, b and c are coplanar. A unit vector d is perpendicular to them. If and the angle between a and b is 30o, then c is (a) (b) (c) (d) 412. If vectors a, b, c satisfy the condition , then is equal to (a) 0 (b) – 1 (c) 1 (d) 2 413. Let r be a vector perpendicular to , where . If , then is (a) 2 (b) 1 (c) 0 (d) None of these 414. Let a, b and c be three vectors having magnitudes 1, 1 and 2 respectively. If , the acute angle between a and c is (a) (b) (c) (d) None of these 415. If and c is a unit vector perpendicular to the vector a and coplanar with a and b, then a unit vector d perpendicular to both a and c is (a) (b) (c) (d) 416. If a is perpendicular to b and r is a non-zero vector such that , then r = (a) (b) (c) (d) 417. Given three vectors a, b, c such that . The vector r which satisfies and is (a) (b) (c) (d) None of these 418. If and are two vectors ,then the point of intersection of two lines and is (a) (b) (c) (d) 419. A line passes through the points whose position vectors are and . The position vector of a point on it at a unit distance from the first point is (a) (b) (c) (d) None of these 420. The projection of the vector on the line whose vector equation is , t being the scalar parameter, is (a) (b) 6 (c) (d) None of these 421. If and , then [a b c] is equal to (a) 0 (b) 1 (c) 2 (d) None of these 422. If and then (a) (b) (c) (d) 423. If r satisfies the equation , then for any scalar m, r is equal to (a) (b) (c) (d) 424. If and , then the vector satisfying the conditions (i) That it is coplanar with a and b (ii) That it is perpendicular to b (iii) That is (a) (b) (c) (d) None of these 425. If the non-zero vectors a and b are perpendicular to each other, then the solution of the equation, is given by (a) (b) (c) (d)