Vector Algebra

Let a = i + j + k, b = 3 i + 2 j - k, c = j + k and d be a unit vector such that a d = b d and c d = 1. If c is perpendicular to a, then |3 d + c|2 is equal to

Let a = i + 2 j + k, b = 3 i - 3 j + 3 k, c = 2 i - j + 2 k and d be a vector such that b d = c d and a d = 4. Then |( a d)|2 is equal to .

Let ABCD be a tetrahedron such that the edges AB, AC and AD are mutually perpendicular. Let the areas of the triangles ABC, ACD and ADB be 5, 6 and 7 square units respectively. Then the area (in square units) of the BCD is equal to :

1. 110
2. 12
3. 340
4. 7 3

If a is a nonzero vector such that its projections on the vectors 2 i - j + 2 k, i + 2 j - 2 k and k are equal, then a unit vector along a is :

1. 1 155(-7 i + 9 j + 5 k)
2. 1 155(-7 i + 9 j - 5 k)
3. 1 155(7 i + 9 j - 5 k)
4. 1 155(7 i + 9 j + 5 k)

Let a = 2 i - 3 j + k, b = 3 i + 2 j + 5 k and a vector c be such that ( a - c) b = -18 i - 3 j + 12 k and a c = 3. If b c = d, then | a d| is equal to :

1. 15
2. 18
3. 12
4. 9

Let a = 2 i - j + 3 k, b = 3 i - 5 j + k and c be a vector such that a c = c b and ( a + c) ( b + c) = 168. Then the maximum value of | c|2 is :

1. 77
2. 154
3. 308
4. 462

Let a = i + j + k, b = 2 i + 2 j + k and d = a b. If c is a vector such that a c = | c|, | c - 2 a|2 = 8 and the angle between d and c is 4, then |10 - 3 b c| + | d c|2 is equal to .

If the components of a = i + j + k along and perpendicular to b = 3 i + j - k respectively, are 1611(3 i + j - k) and 111(-4 i - 5 j - 17 k), then 2 + 2 + 2 is equal to:

1. 16
2. 23
3. 26
4. 18

Let A, B, C be three points in xy-plane, whose position vectors are given by 3 i + j, i + 3 j and a i + (1 - a) j respectively with respect to the origin O. If the distance of the point C from the line bisecting the angle between the vectors OA and OB is 9 2, then the sum of all the possible values of a is:

1. 2
2. 0
3. 92
4. 1