Let  be unit vectors such that . Which one of the following is correct

a)

b)

c)

d)  are mutually perpendicular

Multiple choice

Which of the following expressions are meaningful

a)

b)

c)

d)

Multiple choice

Let a and b be two non-collinear unit vectors. If  and  then  is

a) ||

b)

c)

d)

Find all values of λ such that  and

 where  are unit vectors along the coordinate vectors.

If c be a given non-zero scalar and A and B be given non-zero vectors such that A is perpendicular to B, find the vector X which satisfies the equations

The position vectors of the point A, B, C, D are  respectively. If the points A, B, C and D lie in a plane, find the value of λ.

Let OABC be a parallelogram with O as the origin and OC a diagonal. Let D be the midpoint of OA. Using vector method, prove that BD and CO intersect in the same ratio.

If a, b, c are coplanar, show that
  

Let A = . Determine a vector R satisfying  and .

Determine the value of c so that for all real x the vector cx and  make an obtuse angle with each other.

If a, b, c, d are distinct vectors satisfying relation  and . Prove that

Fill in the blank

Let A, B, C be three vectors of length 3, 4, 5 respectively. Let A is perpendicular to B + C, B is perpendicular to C + A, and C is perpendicular to A + B then the length of the vector  is equal to  . . . .

A, B, C , D are four points in a plane with position vectors a, b, c, d respectively, such that . The point D then is the  . . . . . . .  of the triangle ABC.

If A, B, C are three non-coplanar vectors then
 

If  are given vectors then the vector B satisfying the equation  and  is . . . . .

If the vectors
 

are coplanar then the value of  . . . . . .

The projection of a vector a along and perpendicular to a non-zero vector  are . . . . . and . . . . . respectively.

Given that a = (1, 1, 1), c = (0, 1, −1), a . b = 3,  then b is equal to

A unit vector coplanar with  and  and perpendicular to  is . . . . .

A unit vector perpendicular to the plane determined by the points P (1, -1, 2), Q (2, 0, -1) and R (0, 2, 1) is .  .  .  .  .

A non-zero vector a is parallel to the line of intersection of the plane determined by the vectors  and the plane determined by the vectors . The angle between a and  is .  .  .  .  .

Let a, b and c be three vectors having magnitudes 1, 1 and 2 respectively. If  then the acute angle between a and c is  .  .  .  .  .

Solve