Vector Algebra

If a= i+ j+ k, b= j- k and c be three vectors such that a c= b and a c=3, then c ( a-2 b ) is equal to .

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 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 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 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 a = i + j + k, b = 2 i + 2 j + k and d = a b. If c is a vector such that a c = | c|, | c - 2 a|2 = 8 and the angle between d and c is 4, then |10 - 3 b c| + | d c|2 is equal to .

Let c be the projection vector of b= i+4 k, >0, on the vector a= i+2 j+2 k. If | a+ c|=7, then the area of the parallelogram formed by the vectors b and c is .