Vector Algebra

Let the position vectors of the vertices A, B and C of a tetrahedron ABCD be i+2 j+ k, i+3 j-2 k and 2 i+ j- k respectively. The altitude from the vertex D to the opposite face ABC meets the median line segment through A of the triangle ABC at the point E. If the length of AD is 1103 and the volume of the tetrahedron is 8056 2, then the position vector of E is

1. 16 (7 i+12 j+ k )
2. 112 (7 i+4 j+3 k )
3. 16 (12 i+12 j+ k )
4. 12 ( i+4 j+7 k )

Let the arc AC of a circle subtend a right angle at the centre O. If the point B on the arc AC, divides the arc AC such that length of arc ABlength of arc BC= 15, and OC= OA+ OB, then + 2( 3-1) is equal to

1. 2 3
2. 5 3
3. 2+ 3
4. 2- 3

Let the point A divide the line segment joining the points P(-1,-1,2) and Q(5,5,10) internally in the ratio r:1 (r>0). If O is the origin and ( OQ OA )- 15 | OP OA |2=10, then the value of r is :

1. 7
2. 14
3. 7
4. 3

Let c be the projection vector of b= i+4 k, >0, on the vector a= i+2 j+2 k. If | a+ c|=7, then the area of the parallelogram formed by the vectors b and c is .

Let a and b be two unit vectors such that the angle between them is 3. If a+2 b and 3 a- b are perpendicular to each other, then the number of values of in [-1,3] is :

1. 1
2. 3
3. 2
4. 0