Vector Algebra · Question 10

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

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