Two vectors are given as a = i + 2 j - k and b = 3 i - j + 2 k . A third vector c satisfies the relation 2 a + c = b . The cosine of the angle made by the vector c with the positive z-axis is:

Step-by-step solution

Given the vector equation 2 a + c = b , we can solve for c : c = b - 2 a Substituting the given vectors a and b : c = (3 i - j + 2 k ) - 2( i + 2 j - k ) c = (3 - 2) i + (-1 - 4) j + (2 - (-2)) k c = i - 5 j + 4 k The magnitude of vector c is: | c | = 1^2 + (-5)^2 + 4^2 = 1 + 25 + 16 = 42 The cosine of the angle made by vector c with the positive z-axis is given by the ratio of its z-component to its magnitude: = c_z | c | = 4 42 Answer: 4 42