Vector Algebra

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 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 = 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 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 + 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

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)

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 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

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 = 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

Let a unit vector which makes an angle of 60 with 2 i+2 j- k and an angle of 45 with i- k be C. Then C+ (- 12 i+ 13 2 j- 23 k ) is:

1. - 23 i+ 23 j+ ( 12+ 2 23 ) k
2. ( 1 3+ 12 ) i+ ( 1 3- 13 2 ) j+ ( 1 3+ 23 ) k
3. 23 i- 12 k
4. 23 i+ 13 2 j- 12 k

For > 0, let be the angle between the vectors a = i + j - 3 k and b = 3 i - j + 2 k. If the vectors a + b and a - b are mutually perpendicular, then the value of (14 )2 is equal to

1. 25
2. 50
3. 20
4. 40

Consider a ABC where A(1,3,2), B(-2,8,0) and C(3,6,7). If the angle bisector of BAC meets the line BC at D, then the length of the projection of the vector AD on the vector AC is :

1. 372 38
2. 19
3. 392 38
4. 382

Let a unit vector u=x i+y j+z k make angles 2, 3 and 2 3 with the vectors 1 2 i+ 1 2 k, 1 2 j+ 1 2 k and 1 2 i+ 1 2 j respectively. If v= 1 2 i+ 1 2 j+ 1 2 k then | u- v|2 is equal to

1. 112
2. 52
3. 7
4. 9

The least positive integral value of , for which the angle between the vectors i-2 j+2 k and i+2 j-2 k is acute, is .