Vector Algebra

Consider three vectors a, b, c. Let | a| = 2, | b| = 3 and a = b c. If [0, 3 ] is the angle between the vectors b and c, then the minimum value of 27| c - a|2 is equal to:

1. 124
2. 110
3. 121
4. 105

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

Let ABC be a triangle of area 15 2 and the vectors AB= i+2 j-7 k, BC=a i+b j+c k and AC=6 i+d j-2 k, d>0. Then the square of the length of the largest side of the triangle ABC is .

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

Let a = i + j + k, b = 2 i + 4 j - 5 k and c = x i + 2 j + 3 k, x R. If d is the unit vector in the direction of b + c such that a d = 1, then ( a b) c is equal to

1. 3
2. 9
3. 11
4. 6

Let a=-5 i+ j-3 k, b= i+2 j-4 k and c=((( a b) i) i) i. Then c (- i+ j+ k) is equal to :

1. -12
2. -10
3. -13
4. -15

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= i+ j+ k, b=- i-8 j+2 k and c=4 i+c2 j+c3 k be three vectors such that b a= c a. If the angle between the vector c and the vector 3 i+4 j+ k is , then the greatest integer less than or equal to 2 is .

Let a=3 i+ j-2 k, b=4 i+ j+7 k and c= i-3 j+4 k be three vectors. If a vector p satisfies p b= c b and p a=0, then p ( i- j- k) is equal to

1. 24
2. 32
3. 36
4. 28

Let a and b be two vectors such that | a|=1, | b|=4, and a b=2. If c=(2 a b)-3 b and the angle between b and c is , then 192 2 is equal to .