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An example would be three corners being the points: (1,1), (4,2) and (1,3). I understand the specific solution for this example: (4,4), (4,0) or (-2,2). Which I reasoned when i drew it out. The example came from a linear algebra textbook and i'm curious if it can be done some other way.

Drew Verlee
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The three possibilities come from adding the coordinates of two points and subtracting the third. So $(4,0)=(4,2)+(1,1)-(1,3)$. There are three choices of which point to subtract. You are translating the point you subtract to the origin, then adding the vectors to the other two points to find the opposite one, then translating back. So you could look at it as $(4,0)=[(4,2)-(1,3)]+[(1,1)-(1,3)]+(1,3)$

Ross Millikan
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  • Hi, mr. @Ross Millikan! Would you mind explaining the reasoning behind adding two vectors and subtracting a third from the sum, please? – Eduardo dos Santos Almeida Nov 26 '23 at 12:40
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    @EduardodosSantosAlmeida: in a parallelogram if one vertex is at the origin the opposite point is at the sum of the other two corners. I suggest you draw this out on graph paper-plot the three points and a new set of axes going through $(1,3)$. If you add the two vectors from $(1,3)$ to the other two points you get the fourth. The last expression shows the coordinates in the shifted frame. – Ross Millikan Nov 26 '23 at 14:42
  • Now, I got it! Thank you for your time, mr. @Ross Millikan! – Eduardo dos Santos Almeida Dec 01 '23 at 18:01
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Given 3 vertices, these vertices define 2 vectors; the unknown endpoint corresponds to the sum of these 2 vectors.

Example: $(1,1)$ and $(1,3)$ form the vector $(0,2)$. $(1,1)$ and $(4,2)$ form the vector $(3,1)$. Sum these vectors to get $(3,3)$. Then add the vertex $(1,1)$ back to get $(4,4)$ as the formerly unknown corner.

Ron Gordon
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let the vertices of parallelogram be vectors a,b,c,d where a = (1,1); b = (4,2), c = (1,3), and let d = (x,y). Now for parallelogram there can be three possibility i.e. a - b || c - d or a - c || d - b or a - d || b - c (as opposite sides are parallel) solving from above three statements and equating respective components on LHS and RHS. we can have (x,y) = (4,4) or (4,0) or (-2,1)

vivek upadhyay
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