Let S be the set of all vectors v = x i + y j + z k where x, y, z are integers, such that | v | = 3 (|x|, |y|, |z|) and 0 < | v | 10 . The number of vectors in the set S is

Step-by-step solution

Let k = (|x|, |y|, |z|) . Then k 0 and k is an integer. The magnitude of the vector is given by | v |^2 = x^2 + y^2 + z^2 . According to the given condition, | v |^2 = 3k^2 . Since x^2 k^2 , y^2 k^2 , and z^2 k^2 , the maximum possible value for x^2 + y^2 + z^2 is 3k^2 . This maximum is achieved if and only if x^2 = y^2 = z^2 = k^2 . Thus, |x| = |y| = |z| = k . We are given 0 Solving for k , we get 0 Since k is a positive integer, the possible values for k are 1, 2, 3, 4, 5 . For each value of k , each of the compo