Which of the following point is on the line passes through \(\mathrm{A}(1,2,0)\) and \(\mathrm{B}(3,1,1) ?\) (A) \((7,-1,3)\) (B) \((-7,1,3)\) (C) \((7,-1,-3)\) (D) \((7,1,3)\)

To find the direction vector of the line AB, we need to find the difference between the coordinates of points A and B. Let the direction vector be represented as \(\mathrm{D}\), then: \[ \mathrm{D} = \mathrm{B} - \mathrm{A} = (3-1, 1-2, 1-0) = (2, -1, 1) \]

We will check each option to see if it is on the line passing through A and B. A point is on the line if and only if the difference vector between the given point and A (or B) is parallel to the direction vector D mentioned above. Two vectors are parallel if they are scalar multiples of each other. (A) Check point (7, -1, 3): \[ \mathrm{P_A} = (7-1, -1-2, 3-0) = (6, -3, 3) \] Is \(P_A\) parallel to D? \[ \frac{6}{2} = 3,\ \frac{-3}{-1} = 3,\ \frac{3}{1} = 3 \] Since all ratios are equal, point (7, -1, 3) lies on the line passing through A and B. (B) Check point (-7,1,3): \[ \mathrm{P_A} = (-7-1, 1-2, 3-0) = (-8,-1,3) \] Is \(P_A\) parallel to D? \[ \frac{-8}{2} = -4,\ \frac{-1}{-1} = 1,\ \frac{3}{1} = 3 \] Since all ratios are not equal, point (-7,1,3) does not lie on the line passing through A and B. (C) Check point (7,-1,-3): \[ \mathrm{P_A} = (7-1,-1-2,-3-0) = (6,-3,-3) \] Is \(P_A\) parallel to D? \[ \frac{6}{2} = 3,\ \frac{-3}{-1} = 3,\ \frac{-3}{1} = -3 \] Since all ratios are not equal, point (7,-1,-3) does not lie on the line passing through A and B. (D) Check point (7,1,3): \[ \mathrm{P_A} = (7-1,1-2,3-0) = (6,-1,3) \] Is \(P_A\) parallel to D? \[ \frac{6}{2} = 3,\ \frac{-1}{-1} = 1,\ \frac{3}{1} = 3 \] Since all ratios are not equal, point (7,1,3) does not lie on the line passing through A and B.

As we have checked all the given points, only point (A) (7, -1, 3) is on the line passing through A(1, 2, 0) and B(3, 1, 1).