Addition of Radius-Vectors

Fix a point O in the plane. Point O is called the origin. The directed segment OA from the origin to an arbitrary point A in the plane is known as the A's radius-vector. Radius-vectors of two points can be added according to the rule of parallelogram. Sometimes we forget to mention the origin and talk of the sum A + B of two points, which usually happens in affine geometry. The reason for this laxity is that the sums OA + OB and O'A + O'B are related in a very simple manner. What is the relationship?

 

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The sum A + B is translated in the direction opposite to that of OO', but by the same distance.

The point is to prove that the quadrilateral OS'SO' is a parallelogram. To this end, consider two parallelograms, OASB and O'AS'B. The two share the diagonal AB. As is well known, the parallelogram is characterized by the property that its diagonals are divided in half by the point of their intersection. It thus follows that the diagonals OS and O'S' of the parallelograms OASB and O'AS'B both path through the midpoint of AB and are divided in half by that point.

In other words, in the quadrilateral OS'SO', the diagonals are divided into half by their point of intersection. Therefore, the quadrilateral is a parallelogram.

There's a less formal explanation. Since it's all about vectors, it seems intuitively clear that when one of the points A or B is translated by a vector v, the sum A + B undergoes the same transformation: (A + v) + B = v + (A + B). If both A and B are shifted by v, the sum is translated twice as far, by 2v. However, when three points A, B and O are translated by v, the sum is obviously translated in the same manner, i.e. only by v. Comparing the last two cases we concluded that shifting the origin has a "detrimental" effect on the translation of the sum: instead of moving by 2v, the sum only moves v. It also clear that effects of translation of any of the three points is independent of effects caused by translations of the other two points. Therefore, v - 2v is the effect on the sum A + B of translating the origin by the vector v.

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