Vector Algebra

Let a=2 i+3 j+3 k and b=6 i+3 j+3 k. Then the square of the area of the triangle with adjacent sides determined by the vectors (2 a+3 b ) and ( a- b ) is :

1. 450
2. 900
3. 1800
4. 2400

Let u and v be unit vectors inclined at an acute angle such that | u v |= 32. If A= u+ v+ ( u v ), then is equal to :

1. 43 ( A u )- 23 ( A v )
2. 23 ( A u )- 13 ( A v )
3. 43 ( A u )+ 23 ( A v )
4. ( A u )- 12 ( A v )

Let P be a point in the plane of the vectors AB = 3 i + j - k and AC = i - j + 3 k such that P is equidistant from the lines AB and AC. If | AP| = 52, then the area of the triangle ABP is :

1. 32
2. 304
3. 264
4. 2

Let a = 2 i + j - 2 k, b = i + j and c = a b. Let d be a vector such that | d - a| = 11, | c d| = 3 and the angle between c and d is 4. Then a d is equal to

1. 0
2. 1
3. 3
4. 11

Let a = 2 i - 5 j + 5 k and b = i - j + 3 k. If c is a vector such that 2( a c) + 3( b c) = 0 and ( a - b) c = -97, then | c k|2 is equal to

1. 193
2. 205
3. 218
4. 233

Let a = - i + j + 2 k, b = i - j - 3 k, c = a b and d = c a. Then ( a - b ) d is equal to :

1. 2
2. -2
3. 4
4. -4

Let a, b, c be three vectors such that a b = 2( a c). If | a| = 1, | b| = 4, | c| = 2, and the angle between b and c is 60 , then | a c| is equal to

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

Let a = 2 i - j + k and b = j + 2 k, Z be two vectors. Let c = a b and d be a vector of magnitude 2 in yz-plane. If | c| = 53, then the maximum possible value of ( c d)2 is equal to :

1. 52
2. 26
3. 104
4. 208

Let c and d be vectors such that | c + d| = 29 and c (2 i + 3 j + 4 k) = (2 i + 3 j + 4 k) d. If 1, 2 ( 1 > 2) are the possible values of ( c + d) (-7 i + 2 j + 3 k), then the equation K2x2 + (K2 - 5K + 1)xy + (3K + 22 )y2 - 8x + 12y + 2 = 0 represents a circle, for K equal to :

1. 4
2. -1
3. 2
4. 1

For a triangle ABC, let p = BC, q = CA and r = BA. If | p| = 2 3, | q| = 2 and = 1 3, where is the angle between p and q, then | p ( q - 3 r)|2 + 3| r|2 is equal to :

1. 200
2. 220
3. 410
4. 340

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

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 be a unit vector perpendicular to the vectors b = i - 2 j + 3 k and c = 2 i + 3 j - k, and a makes an angle of -1 (- 13 ) with the vector i + j + k. If a makes an angle of 3 with the vector i + j + k, then the value of is:

1. 3
2. 6
3. - 6
4. - 3

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 .

Let a=3 i- j+2 k, b= a ( i-2 k) and c= b k. Then the projection of c-2 j on a is :

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

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 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 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

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

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 = 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