Vector Algebra · Question 5

Let ak=( k) i+ j and bk= i-( k) j, where k= 2k-1 2n+1, for some n N, n>5. Then the value of k=1n | ak |2 k=1n | bk |2 is

Let u and v be unit vectors inclined at an acute angle such that | u v |= 32. If A= u+ v+ ( u v ), then is equal to :

1. 43 ( A u )- 23 ( A v )
2. 23 ( A u )- 13 ( A v )
3. 43 ( A u )+ 23 ( A v )
4. ( A u )- 12 ( A v )

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 PQR be a triangle such that PQ = -2 i - j + 2 k and PR = a i + b j - 4 k, a, b Z. Let S be the point on QR, which is equidistant from the lines PQ and PR. If | PR| = 9 and PS is parallel to i - 7 j + 2 k, then the value of 3a - 4b is .

Let P be a point in the plane of the vectors AB = 3 i + j - k and AC = i - j + 3 k such that P is equidistant from the lines AB and AC. If | AP| = 52, then the area of the triangle ABP is :

1. 32
2. 304
3. 264
4. 2

Let a = 2 i + j - 2 k, b = i + j and c = a b. Let d be a vector such that | d - a| = 11, | c d| = 3 and the angle between c and d is 4. Then a d is equal to

1. 0
2. 1
3. 3
4. 11

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