Vector Question

Question

A rectangle R³ has three of its corners at the points (3,1,1), (2,3,4) and (1,1,2). What are the coordinates of its fourth corner?

Answer 1

Let's say the corners are labeled as A=(2,3,4), B=(1,1,2), C=(3,1,1), and D=(x,y,z).

And you know in each rectangle, diameter > length> width.

As |AC|=sqrt{14} > |AB|= 3 > |BC|=sqrt{5} so we get that the rectangle is of the form ABCD( the vector AC is one of the diameters, the vector AB is one of the side vectors, and the vector BC is the other side).

From the form the rectangle, we can see AB and DC are parallel, so they are the same vector, I hope this part makes sense to you, you may need to visualize it.

AB = B-A= (-1,-2,-2), DC = C-D= (3-x,1-y,1-z), so setting these two vectors equal you get D=(4, 3,3).