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 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 .

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

Let a= i+2 j+3 k, b=3 i+ j- k and c be three vectors such that c is coplanar with a and b. If the vector c is perpendicular to b and a c=5, then | c| is equal to

1. 116
2. 13 2
3. 18
4. 16

Let the position vectors of three vertices of a triangle be 4 p+ q-3 r, -5 p+ q+2 r and 2 p- q+2 r. If the position vectors of the orthocenter and the circumcenter of the triangle are p+ q+ r4 and p+ q+ r respectively, then +2 +5 is equal to :

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

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 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