Fourth point of a parallelogram with corners at $\langle 1,1\rangle$, $\langle 4,2\rangle$ and $\langle 1,3\rangle$? I have the three position vectors for the corners of a parallelogram: $\langle 1,1\rangle$, $\langle 4,2\rangle$ and $\langle 1,3\rangle$.
The question is asking for the three possible candidates for the fourth point.
I drew the three points and deduced that the three candidates are $(-2,2)$, $(4,4)$ and $(4,0)$.
My question is: Is there a general way to deduce the points from the vectors without drawing a picture? I find it kinda hard since I need to know how the points are aligned.
 A: Yes.  There is a general way.  Take the points in sequence (first, second, third), and compute the two vectors from the point that you selected, to the remaining two points, then the fourth vertex is the point that you selected plus the sum the the two vectors that you computed.
So, starting with $\langle 1,1 \rangle$, the two sides vectors are $\langle 3, 1 \rangle $ and $ \langle 0, 2 \rangle $, so the fourth vertex in this case will be $ \langle 4, 4 \rangle $
Now, take the second point $ \langle 4,2 \rangle $ then the two sides vectors will be $ \langle -3, -1 \rangle $ and $  \langle -3, 1 \rangle $ , so the fourth vertex will be $ \langle -2 , 2 \rangle $
Finally, take the third point $ \langle 1, 3 \rangle $ then the two sides vectors will be $ \langle 0, -2 \rangle$ and $ \langle 3, -1 \rangle$ , so the fourth vertex will be $ \langle 4, 0 \rangle $
A: There is another answer to this question here, but I'll provide a little bit of intuition so that, hopefully, the formula will make sense.
You already know the three points provided, $P$, $Q$, and $R$, will form one vertex of the parallelogram. The only question now is which pairs of vertices form sides. Fortunately, there are only three pairs to choose from: $P$ & $Q$, $Q$ & $R$, and $R$ & $P$.
Since we want another side that is congruent and parallel to one of the sides given, we can essentially "copy and paste" one of the sides. To do that, we'll take the difference between two vectors to get the "change" required to form a congruent, parallel segment and then add that "change" to the third vector.
Using the example you've provided, we'll start with the pair $\langle1,1\rangle$ and  $\langle4,2\rangle$. Taking the difference, we get $\langle-3,-1\rangle$, and adding it to $\langle1,3\rangle$, we get $\langle-2,2\rangle$. Doing the same with the next two, we get $\langle4,0\rangle$, and with the last pair, we get $\langle4,4\rangle$.
The answer I linked does mention six, but as it says there, the other three are repeats. I'll leave it up to you as an extra problem to think about why this method works without determining the fourth side for each pair.
