Vector Algebra

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

Let a = 6 i + j - k and b = i + j. If c is a vector such that | c| 6, a c = 6| c|, | c - a| = 2 2 and the angle between a b and c is 60 , then |( a b) c| is equal to:

1. 32 6
2. 92(6 - 6)
3. 92(6 + 6)
4. 32 3

If A(1, -1, 2), B(5, 7, -6), C(3, 4, -10) and D(-1, -4, -2) are the vertices of a quadrilateral ABCD, then its area is :

1. 24 7
2. 48 7
3. 24 29
4. 12 29