Vector Algebra · Question 15

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