The rectangle ABCD has vertices at A = (1,2,3), B = (3,6,-2) and C = (0,5,-4). D

Question

The rectangle ABCD has vertices at A = (1,2,3), B = (3,6,-2) and C = (0,5,-4). Determine the coordinates of vertex D.

Explanation / Answer

Dear Anonymous, The main idea behind this problem is that a rectangle is a parallelogram. We know that the vertices are in the order of ABCD, and furthermore, we know three of the vertices. Therefore, because of the fact that the rectangle is a parallelogram, we know that the vector that goes from A to B, let us call it AB, will be parallel to and equal in magnitude to the vector that goes from C to D. Similarly the vector From B to C, let us call it BC will be equal in magnitude and parallel to AD. Therefore, I am going to calculate the vector that goes from B to A, call it BA as follows: BA = A - B = (1, 2, 3) - (3, 6, -2) = < -2, -4, 5>. I have used the brackets simply to emphasize that BA is a vector. From the previous argument, I know that BA will be parallel and of the same magnitude as CD; additionally, if you draw the rectangle, you will see that BA and CD are both going in the same direction. Thus in order to locate D, I am going to place the beginning endpoint of BA on C, and find out where the tip is: this tip will be the location of D. This is done mathematically by taking the coordinates of C and adding those of BA as shown: D = BA + C = (-2 +0, -4+5, 5+(-4)) = ( -2, 1, 1). Thus we have located the coordinates of the vertex D. Intuitively, you can imagine that you were standing on C, and that you walked 2 steps in the negative x direction, 4 steps in the negative y direction, and 5 steps in the positive z direction: this is one physical interpretation of a vector. I hope that this helps!!! David

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