Addition of points in the plane
Coordinate Definition (Coordinate Method)
The sum of two points with the rectangular coordinates [a,b] and [c,d] is given by [a+c,b+d]. We therefore write
For example [2,5]+ [6,2] = [8,7].
[a,b] + [c,d] = [a+c,b+d]
Associative and commutative Axioms for real numbers imply addition of points in the plane is associative and commutative.
In words, the addition rule is simple add the rectangular coordinates of the summands to get the rectangular coordinates of the sum. With this in mind, the following question is easy: What are the rectangular coordinates of the sum of [1,14] and [2,8]? Answer:
[1,14]+ [2,8] = [1+2,14+8] = [3,22].
The Addition Parallelogram
Assumption: The origin [0,0], the two points [a,b] and [c,d] are not collinear:
The origin [0,0], the two points [a,b] and [c,d], and their [a,b] +[c,d] provide the vertices of a quadrilateral in which opposite sides have equal lengths. The proof follows.
Here the distance between [a,b] and [a,b] +[c,d] = [a+c,b+d] is the same as the distance between [c,d] and [0,0] by the isometry of two right triangles with hypotenuses respectively given by the line segments
[0,0] to [c,d] (side 0P)
and
[a,b] to [a,b] +[c,d] = [a+c,b+d] (side QS)
Whence one pair of opposite sides OP and QS in the quadrilateral OPSQ have equal lengths. Likewise, the sides OQ and PS form a second pair of opposite sides with equal lengths. Thus the quadrilateral is a parallelogram.
Conclusion: The addition of points [a,b] and [c,d] not collinear with the origin [0,0] yields a point [a+c,b+d] with the property that the origin [0,0], the two points [a,b] and [c,d], and their sum [a+c,b+d] form the vertices of parallelogram with the line segment (i) [0,0] to [a, b], and (ii) [0,0] to [c,d] being adjacent.
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