Vector Algebra

Let a=4 i- j+3 k, b=10 i+2 j- k and a vector c be such that 2 ( a b )+3 ( b c )= 0. If a c=15, then c ( i+ j-3 k ) is equal to :

1. -6
2. -5
3. -4
4. -3

If a= i+ j+ k, b= j- k and c be three vectors such that a c= b and a c=3, then c ( a-2 b ) is equal to .

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 a= 7 i+ j- k and b= j+2 k. If r is a vector such that r a+ a b= 0 and r a=0, then |3 r|2 is equal to:

1. 44
2. 54
3. 86
4. 132

Let O be the origin, OP= a and OQ= b. If R is the point on OP such that OP=5 OR, and M is the point such that OQ=5 RM, then PM is equal to :

1. 15 ( a-4 b )
2. 15 ( b-4 a )
3. 15 (- a+4 b )
4. 15 (- b+4 a )

Let ak=( k) i+ j and bk= i-( k) j, where k= 2k-1 2n+1, for some n N, n>5. Then the value of k=1n | ak |2 k=1n | bk |2 is

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 )

If a and b are two vectors such that | a|=2 and | b|=3, then the maximum value of 3 | (3 a+2 b ) |+4 | (3 a-2 b ) | is :

1. 30
2. 36
3. 60
4. 72

Let the vectors a=- i+ j+3 k and b= i+3 j+ k. For some , R, let c= a+ b. If c (3 i-6 j+2 k )=10 and c ( i+ j+ k )=-2, then | c|2 is equal to :

1. 8
2. 12
3. 14
4. 15

Two adjacent sides of a parallelogram PQRS are given by PQ= j+ k and PS= i- j. If the side PS is rotated about the point P by an acute angle in the plane of the parallelogram so that it becomes perpendicular to the side PQ, then 2 ( 5 2 )- 2 ( 2 ) is equal to:

1. 12
2. 32
3. 34
4. 2 35

For three unit vectors a, b, c satisfying | a - b|2 + | b - c|2 + | c - a|2 = 9 and |2 a + k b + k c| = 3, the positive value of k is :

1. 4
2. 5
3. 6
4. 3

Let PQR be a triangle such that PQ = -2 i - j + 2 k and PR = a i + b j - 4 k, a, b Z. Let S be the point on QR, which is equidistant from the lines PQ and PR. If | PR| = 9 and PS is parallel to i - 7 j + 2 k, then the value of 3a - 4b is .

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 - j - k, b = i + 3 j - k and c = 2 i + j + 3 k. Let v = a + b, R, be the vector in the plane of the vectors a and b, such that the length of its projection on the vector c is 1 14. Then | v| is equal to

1. 352
2. 13
3. 212
4. 7

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 = i - 2 j + 3 k, b = 2 i + j - k, c = i + j + k and v = a b. If v c = 11 and the length of the projection of b on c is p, then 9p2 is equal to

1. 9
2. 6
3. 4
4. 12

Let AB = 2 i + 4 j - 5 k and AD = i + 2 j + k, R. Let the projection of the vector v = i + j + k on the diagonal AC of the parallelogram ABCD be of length one unit. If , , where > , be the roots of the equation 2x2 - 6 x + 5 = 0, then 2 - is equal to

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

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 a vector a = 2 i - j + k, > 0, make an obtuse angle with the vector b = - 2 i + 4 2 j + 4 2 k and an angle , 6 < < 2, with the positive z-axis. If the set of all possible values of is ( , ) - \ \, then + + is equal to .

Let a = - i + 2 j + 2 k, b = 8 i + 7 j - 3 k and c be a vector such that a c = b. If c ( i + j + k) = 4, then | a + c|2 is equal to :

1. 30
2. 33
3. 27
4. 35

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 + 2 j + k and b = 2 i + j - k. Let c be a unit vector in the plane of the vectors a and b and be perpendicular to a. Then such a vector c is:

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

Let the angle , 0 < < 2 between two unit vectors a and b be -1 ( 659 ). If the vector c = 3 a + 6 b + 9( a b), then the value of 9( c a) - 3( c b) is

1. 31
2. 29
3. 24
4. 27

Let a and b be the vectors of the same magnitude such that | a + b| + | a - b|| a + b| - | a - b| = 2 + 1. Then | a + b|2| a|2 is :

1. 2 + 2
2. 2 + 4 2
3. 4 + 2 2
4. 1 + 2

Consider two vectors u = 3 i - j and v = 2 i + j - k, > 0. The angle between them is given by -1 ( 52 7 ). Let v = v1 + v2, where v1 is parallel to u and v2 is perpendicular to u. Then the value of | v1 |2 + | v2 |2 is equal to

1. 232
2. 252
3. 10
4. 14

Let the three sides of a triangle ABC be given by the vectors 2 i- j+ k, i-3 j-5 k and 3 i-4 j-4 k. Let G be the centroid of the triangle ABC. Then 6 (| AG|2+| BG|2+| CG|2 ) is equal to .