For three vectors which of the following expressions is not equal to any of the remaining three
a)
b)
c)
d)
Your Answer
Angle between vectors are unit vectors satisfying
If form a triangle then the internal angle of the triangle between a and b is
Let P, Q, R and S be points on the plane with position vectors respectively. The quadrilateral PQRS must be a
a) Parallelogram which is neither a rhombus nor a rectangle
b) Square
c) Rectangle but not a square
d) Rhombus but not a square
The adjacent sides of a parallelogram ABCD are given by and . The side AD is rotated by an angle α in the plane of the parallelogram so that AD become AD´. If AD´ makes a right angle with the side AB, the cosine of the angle α is given by
and are reciprocal systems of vectors, prove that
Let and be three non-zero vectors such that c is a unit vector perpendicular to both the vectors a and b and the angle between the vectors a and b is then is equal to
a) 1
d) None of these
A vector a has components 2p and 1 with respect to a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If with respect to new system a has components p + 1 and 1 then
a) p ≠ 0
b) p = 1 or p =
c) p = −1 or
d) p = 1 or p = −1
e) None of these
If are three non–coplanar vectors, then
equals
a) 0
Let p, q, r be three mutually perpendicular vectors of the same magnitude. If x satisfies the equation p ((x – q) p) + q ((x – r) q) + r ((x – p) r) = 0 then x is given by
Let and a unit vector c be coplanar. If c is perpendicular to a then c is equal to
A unit vector which is orthogonal to the vectors and
coplanar with the vectors and is
Multiple choice
Let be three vectors. A vector in the plane of b and c whose projection on a is of magnitude is
The vector is
a) A unit vector
b) Makes an angle with the vector
c) Parallel to vector
d) Perpendicular to the vector
Let A be vector parallel to the line of intersection of planes P1 and P2. Plane P1 is parallel to the vectors and and that P2 is parallel to and , then the angle between vector A and a given vector is
A1, A2, …… , An are the vertices of a regular polygon with n sides and O is the centre. Show that
A vector A has components A1, A2, A3 in a right handed rectangular cartesian coordinate system OXYZ. The coordinate system is rotated about the X–axis through an angle . Find the components of A in the new co-ordinate system in terms of A1, A2, A3.
In a triangle OAB, E is the midpoint of BO and D is a point on AB such that AD : DB = 2 : 1. If OD and AE intercept at P determine the ratio OP : PD using vector methods.
If the vectors b, c, d, are not coplanar then prove that a is parallel to the vector
The position vectors of the vertices A, B, C of a tetrahedron are respectively. The altitude from the vertex D to the opposite face ABC meets the median line through A of the triangle ABC at E. If the length of the side AD is 4 and the volume of the tetrahedron is . Find the position vector of E or all possible positions.
If A, B, C are such that |B| = |C|. Prove that
Let be non–coplanar unit vectors equally inclined to one another at an angle θ. If find p, q, r in terms of θ
Prove by vector method or otherwise, that the point of intersection of the diagonals of a trapezium lies on the line passing through the mid points of the parallel sides (you may assume that the trapezium is not a parallelogram)
For any two vectors u and v prove that
i)
ii)