Vector Algebra

Let a = i - 3 j + 7 k, b = 2 i - j + k and c be a vector such that ( a + 2 b) c = 3( c a). If a c = 130, then b c is equal to .

Let a = 2 i + 5 j - k, b = 2 i - 2 j + 2 k and c be three vectors such that ( c + i) ( a + b + i) = a ( c + i). If a c = -29, then c (-2 i + j + k) is equal to:

1. 15
2. 10
3. 5
4. 12

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 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 .

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 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 .

Let a=a1 i+a2 j+a3 k and b=b1 i+b2 j+b3 k be two vectors such that | a|=1, a b=2 and | b|=4. If c=2( a b)-3 b, then the angle between b and c is equal to:

1. -1 (- 1 3 )
2. -1 ( 23 )
3. -1 ( 2 3 )
4. -1 (- 32 )

Let A(2,3,5) and C(-3,4,-2) be opposite vertices of a parallelogram ABCD. If the diagonal BD= i+2 j+3 k, then the area of the parallelogram is equal to :

1. 12 410
2. 12 306
3. 12 586
4. 12 474

Let a= i+ j+ k, , R. Let a vector b be such that the angle between a and b is 4 and | b|2=6. If a b=3 2, then the value of ( 2+ 2 )| a b|2 is equal to

1. 85
2. 90
3. 75
4. 95

Let a and b be two vectors such that | b|=1 and | b a|=2. Then |( b a)- b|2 is equal to

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

Let OA= a, OB=12 a+4 b and OC= b, where O is the origin. If S is the parallelogram with adjacent sides OA and OC, then area of the quadrilateral OABCarea of S is equal to .

1. 7
2. 6
3. 8
4. 10