Addition of RadiusVectors
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 radiusvector. Radiusvectors of two points can be added according to the rule of parallelogram. Sometimes we forget to mention the origin and talk of the sum

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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
Activities Contact Front page Contents Geometry
Copyright © 19962018 Alexander Bogomolny
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