Let the vectors a = - i + j + 3 k and b = i + 3 j + k . For some , R , let c = a + b . If c (3 i - 6 j + 2 k ) = 10 and c ( i + j + k ) = -2 , then | c |^2 is equal to:

Step-by-step solution

Given a = - i + j + 3 k and b = i + 3 j + k . The vector c is given by: c = a + b = (- + ) i + ( + 3 ) j + (3 + ) k Using the first condition c (3 i - 6 j + 2 k ) = 10 : 3(- + ) - 6( + 3 ) + 2(3 + ) = 10 -3 + 3 - 6 - 18 + 6 + 2 = 10 -3 - 13 = 10 (1) Using the second condition c ( i + j + k ) = -2 : (- + ) + ( + 3 ) + (3 + ) = -2 3 + 5 = -2 (2) Adding equation (1) and equation (2): -8 = 8 = -1 Substituting = -1 into equation (2): 3 + 5(-1) = -2 3 = 3 = 1 Substituting the values of and back into c : c = (-(1) + (-1))