Vector Algebra

Let O be the origin and the position vectors of A and B be 2 i+2 j+ k and 2 i+4 j+4 k respectively. If the internal bisector of meets the line AB at C, then the length of OC is

1. 32 34
2. 23 31
3. 23 34
4. 32 31

Let a unit vector u=x i+y j+z k make angles 2, 3 and 2 3 with the vectors 1 2 i+ 1 2 k, 1 2 j+ 1 2 k and 1 2 i+ 1 2 j respectively. If v= 1 2 i+ 1 2 j+ 1 2 k then | u- v|2 is equal to

1. 112
2. 52
3. 7
4. 9

Let OA= a, OB=12 a+4 b and OC= b, where O is the origin. If S is the parallelogram with adjacent sides OA and OC, then area of the quadrilateral OABCarea of S is equal to .

1. 7
2. 6
3. 8
4. 10

Let a= i+2 j+ k, b=3( i- j+ k). Let c be the vector such that a c= b and a c=3. Then a (( c b)- b- c) is equal to :

1. 32
2. 36
3. 24
4. 20

The least positive integral value of , for which the angle between the vectors i-2 j+2 k and i+2 j-2 k is acute, is .

The position vectors of the vertices A, B and C of a triangle are 2 i-3 j+3 k, 2 i+2 j+3 k and - i+ j+3 k respectively. Let l denote the length of the angle bisector AD of where D is on the line segment BC, then 2l2 equals :

1. 45
2. 50
3. 42
4. 49