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

Let three vectors a = i + 4 j + 2 k, b = 5 i + 3 j + 4 k, c = x i + y j + z k form a triangle such that c = a - b and the area of the triangle is 5 6. If is a positive real number, then | c|2 is equal to:

1. 14
2. 12
3. 16
4. 10

Let OA = 2 a, OB = 6 a + 5 b and OC = 3 b, where O is the origin. If the area of the parallelogram with adjacent sides OA and OC is 15 sq. units, then the area (in sq. units) of the quadrilateral OABC is equal to:

1. 32
2. 38
3. 35
4. 40

Between the following two statements:

1. Both Statement I and Statement II are correct.
2. Both Statement I and Statement II are incorrect.
3. Statement I is correct but Statement II is incorrect.
4. Statement I is incorrect but Statement II is correct.

Let a = 2 i + j + k, b = - i + k, c = j - k, where and are integers and = -6. Let the values of the ordered pair ( , ), for which the area of the parallelogram of diagonals a + b and b + c is 212, be ( 1, 1) and ( 2, 2). Then 12 + 12 - 2 2 is equal to

1. 21
2. 24
3. 19
4. 17

The set of all , for which the vectors a = t i + 6 j - 3 k and b = t i - 2 j - 2 t k are inclined at an obtuse angle for all t R, is

1. [0, 1)
2. (- 43, 0 ]
3. (-2, 0]
4. (- 43, 1 )