Linear Algebra - Question on finding points on plane

Questions
1

“Find a point R in the 3-dimensional space such that the points (0, 0, 0), (1, 2, 3), (2, 2, ?2), and R form a rectangle”

Could somebody please give me some direction as to how I would go about answering this question? Help is greatly appreciated, thanks!

2

The three vectors (better expressed as Mathematica Lists with curly brackets) are the displacements of the three points from the origin. To define a plane the three of them should not be scalar multiples of one another. {0,0,0} is a scalar multiple of the other two but Is the third point displacement a multiple of the second point displacement?

In fact, to be part of a rectangle, the second and third displacements must be at right angles, which means their dot product is zero. Is it?

Then the displacement to the point R would be the sum of the displacements to the second and third point. So what is it?

3

Hi David, thank you for your response.
Could you provide me with your answer to see whether I have arrived at the same conclusion?

4

Jordan, a quick and dirty solution, using a CAD program, gives the point {3, 4, 1}. This is how it looks in Mathematica:

Graphics3D[
 	Polygon[{{0, 0, 0}, {1, 2, 3}, {3, 4, 1}, {2, 2, -2}}],
 	Axes -> True, AxesOrigin -> {0, 0, 0}, Boxed -> False
 ]

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5

And of course David already gave you the solution:

Then the displacement to the point R would be the sum of the
displacements to the second and third point. So what is it?

6

The 4th point would be the sum of the vectors {1,2,3} + {2,2,-1} = {3,4,1}.