The points A(3, -4, 5), B(1, 0, 5) and C(3, 1, -2) are three of four vertices of parallelogram ABCD. What're the three possibilities for the location of D?

Question

Has to do with vectors.

Answer 1

Jamie, I hope I can still help! I had to think long and hard about this one!

 

Connecting our three points, we get a triangle. A triangle is half a parallelogram, which is why the area of a triangle is 1/2 bh. Each side of the triangle acts as a diagonal of the parallelograms.

 

Let's start with AB. It will be the diagonal where BD is opposite AC and BC is opposite AD. A - C = (0,-5,7). Since D - B, sould give us the same result, D = B + (0,-5,7) = (1,-5,12).

 

Now, BC. It's the diagonal of a parallelogram with AC again being opposite of BD but with AB being opposite CD. We again use A - C, and since B - D should give us (0,-5,7), D = B - (0,-5,7) = (1,5,-2).

 

Lastly, the parallelogram with AC as its diagonal will have AB and CD as opposites, and BC and AD. A - B = (2,-4,0). D - C = (2,-4,0), so D = C + (2,-4,0) = (5,-3,-2).

 

Hope this helps!