Vector Algebra

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 )

Let a = 9 i - 13 j + 25 k, b = 3 i + 7 j - 13 k and c = 17 i - 2 j + k be three given vectors. If r is a vector such that r a = ( b + c) a and r ( b - c) = 0, then |593 r + 67 a|2(593)2 is equal to .

Let a = 4 i - j + k, b = 11 i - j + k and c be a vector such that ( a + b) c = c (-2 a + 3 b). If (2 a + 3 b) c = 1670, then | c|2 is equal to:

1. 1600
2. 1618
3. 1627
4. 1609

Let a = i + 2 j + 3 k, b = 2 i + 3 j - 5 k and c = 3 i - j + k be three vectors. Let r be a unit vector along b + c. If r a = 3, then 3 is equal to:

1. 21
2. 25
3. 27
4. 30

If A(3, 1, -1), B ( 53, 73, 13 ), C(2, 2, 1) and D ( 103, 23, -13 ) are the vertices of a quadrilateral ABCD, then its area is

1. 4 23
2. 2 23
3. 5 23
4. 2 2

Let a = 2 i - 3 j + 4 k, b = 3 i + 4 j - 5 k and a vector c be such that a ( b + c ) + b c = i + 8 j + 13 k. If a c = 13, then (24 - b c ) is equal to .

Let a = 2 i + j - k, b = (( a ( i + j)) i) i. Then the square of the projection of a on b is:

1. 13
2. 15
3. 2
4. 23