Final answer:
To find the coordinates of vertex D in the rectangle ABCD, the directed segments AC and AB are computed and AB is added to point C to determine D, resulting in D = (3, 10, -8).
Step-by-step explanation:
The question involves finding the coordinates of the fourth vertex of a rectangle, given the coordinates of the other three vertices. To determine the coordinates of vertex D, one can use the fact that in a rectangle, opposite sides are equal and parallel. Given the vertices A = (1, 2, 3), B = (3, 6, -2), and C = (1, 6, -3), we can find the vector AC and vector AB to help us determine the fourth vertex D. Vector AC is the vector that represents the direction and distance from A to C, and vector AB represents the direction and distance from A to B.
Firstly, we find vector AC by subtracting the coordinates of A from C: AC = C - A = (1 - 1, 6 - 2, -3 - 3) = (0, 4, -6).
Similarly, vector AB is found by subtracting the coordinates of A from B: AB = B - A = (3 - 1, 6 - 2, -2 - 3) = (2, 4, -5).
To find the coordinates of D, we can add vector AB to point C. The coordinates of D are therefore D = C + AB = (1, 6, -3) + (2, 4, -5) = (3, 10, -8).