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

a)

b)

c)

d)

 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

b)

c)

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

b)

c)

d)

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

a)

b)

c)

d)

For three vectors  which of the following expressions is not equal to any of the remaining three

a)

b)

c)

d)

Let and a unit vector c be coplanar. If c is perpendicular to a then c is equal to

a)

b)

c)

d)

A unit vector which is orthogonal to the vectors  and

coplanar with the vectors  and  is

a)

b)

c)

d)

Multiple choice

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

a)

b)

c)

d)

A₁, A₂, …… , A_n 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)