Vector Algebra · Question 23

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

Let a = i + 2 j + k and b = 2 i + j - k. Let c be a unit vector in the plane of the vectors a and b and be perpendicular to a. Then such a vector c is:

1. 1 2(- i + k)
2. 1 5( j - 2 k)
3. 1 3( i - j + k)
4. 1 3(- i + j - k)

Let the angle , 0 < < 2 between two unit vectors a and b be -1 ( 659 ). If the vector c = 3 a + 6 b + 9( a b), then the value of 9( c a) - 3( c b) is

1. 31
2. 29
3. 24
4. 27

Let a and b be the vectors of the same magnitude such that | a + b| + | a - b|| a + b| - | a - b| = 2 + 1. Then | a + b|2| a|2 is :

1. 2 + 2
2. 2 + 4 2
3. 4 + 2 2
4. 1 + 2

Consider two vectors u = 3 i - j and v = 2 i + j - k, > 0. The angle between them is given by -1 ( 52 7 ). Let v = v1 + v2, where v1 is parallel to u and v2 is perpendicular to u. Then the value of | v1 |2 + | v2 |2 is equal to

1. 232
2. 252
3. 10
4. 14

Let the three sides of a triangle ABC be given by the vectors 2 i- j+ k, i-3 j-5 k and 3 i-4 j-4 k. Let G be the centroid of the triangle ABC. Then 6 (| AG|2+| BG|2+| CG|2 ) is equal to .

Let a = i + j + k, b = 3 i + 2 j - k, c = j + k and d be a unit vector such that a d = b d and c d = 1. If c is perpendicular to a, then |3 d + c|2 is equal to

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

Let ABCD be a tetrahedron such that the edges AB, AC and AD are mutually perpendicular. Let the areas of the triangles ABC, ACD and ADB be 5, 6 and 7 square units respectively. Then the area (in square units) of the BCD is equal to :

1. 110
2. 12
3. 340
4. 7 3