16) If \( A, B \) and \( C \) are three point in space, where \( A(3,4,8), B(1,2,7) \) and \( C(3,0,6) \), then \( \qquad \) a) \( \mathbf{2 A B}=\boldsymbol{B C} \) b) \( A B=2 B C \) c) \( A B=B C \) d) \( \mathbf{3 A B}=B C \)

To solve the problem, we need to calculate the vectors \( \mathbf{AB} \) and \( \mathbf{BC} \) based on the given points \( A(3,4,8) \), \( B(1,2,7) \), and \( C(3,0,6) \). 1. **Calculate the vector \( \mathbf{AB} \)**: \[ \mathbf{AB} = B - A = (1 - 3, 2 - 4, 7 - 8) = (-2, -2, -1) \] 2. **Calculate the vector \( \mathbf{BC} \)**: \[ \mathbf{BC} = C - B = (3 - 1, 0 - 2, 6 - 7) = (2, -2, -1) \] Now we have: - \( \mathbf{AB} = (-2, -2, -1) \) - \( \mathbf{BC} = (2, -2, -1) \) Next, we will analyze each option: a) \( \mathbf{2AB} = \mathbf{BC} \) Calculating \( \mathbf{2AB} \): \[ \mathbf{2AB} = 2 \cdot (-2, -2, -1) = (-4, -4, -2) \] This is not equal to \( \mathbf{BC} \). b) \( AB = 2BC \) Calculating \( 2BC \): \[ 2BC = 2 \cdot (2, -2, -1) = (4, -4, -2) \] This is not equal to \( AB \). c) \( AB = BC \) We already have: - \( AB = (-2, -2, -1) \) - \( BC = (2, -2, -1) \) This is not true since \( AB \) and \( BC \) are not equal. d) \( \mathbf{3AB} = \mathbf{BC} \) Calculating \( \mathbf{3AB} \): \[ \mathbf{3AB} = 3 \cdot (-2, -2, -1) = (-6, -6, -3) \] This is not equal to \( \mathbf{BC} \). After evaluating all options, none of the conditions \( a, b, c, \) or \( d \) hold true. Thus, the answer is that none of the statements are correct.