Vector Algebra

Let a = 2 i - 3 j + 4 k, b = 3 i + 4 j - 5 k and a vector c be such that a ( b + c ) + b c = i + 8 j + 13 k. If a c = 13, then (24 - b c ) is equal to .

Let a = 2 i + j - k, b = (( a ( i + j)) i) i. Then the square of the projection of a on b is:

1. 13
2. 15
3. 2
4. 23

Let a = 6 i + j - k and b = i + j. If c is a vector such that | c| 6, a c = 6| c|, | c - a| = 2 2 and the angle between a b and c is 60 , then |( a b) c| is equal to:

1. 32 6
2. 92(6 - 6)
3. 92(6 + 6)
4. 32 3

If A(1, -1, 2), B(5, 7, -6), C(3, 4, -10) and D(-1, -4, -2) are the vertices of a quadrilateral ABCD, then its area is :

1. 24 7
2. 48 7
3. 24 29
4. 12 29

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 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