10.5#19 (edited)

Hello Professor, I've done this problem seven times now and I've gotten various answers, all of which have been wrong. I know I'm using the right process because I've watched seven different videos and they all use the same process. Can you please help me? Even if it's not on time that's okay. Thank you!

(edit) Here are some of the ways I’ve tried to solve it based off of the textbook and videos I’ve watched. Thank you again for your help!

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I'd be happy to help you. Could you do me a favor though, so that I know what I'm helping you with, and email me again with some details about what you tried. For example it would work if you could send me a photo of your calculations if it's not too messy. When I get that I'll edit this post.

(edit) Hmmm, you just got tangled in the arithmetic and subtracted B from A instead of adding when you computed the midpoint, once you made that basic mistake it was hard to spot. It looks like you did the rest correctly...so, when you have your two points A and B, and then some other point C(x,y,z), pictured in red below, which has the same distance 𝑙 from A and B, first of all you have three points so you have a plane. 

And in that plane the three points make an equilateral triangle, so the line that goes from the midpoint M=[(-2,-1,3)+(-1,2,2)]/2=(-3/2,1/2,5/2) of the line between A and B to C must per perpendicular to the line between A and B. And since the midpoint of the line between A and B is equidistant between A and B, is also in that plane of points equidistant from A and B. All this to say you have vec(A,B) ⊥vec(MC) or in otherwords

<-1,-3,1>.<x+3/2,y-1/2,z-5/2>=0= -(x+3/2)-3(y-1/2)+(z-5/2)=-x-3y+z-5/2