Question 1206236
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Geometrically, the sum of two vectors is a DIAGONAL of the parallelogram, built on these vectors as on consecutive sides.


Then the difference of the vectors is the OTHER DIAGONAL of the same parallelogram, built on these vectors as on consecutive sides.


If the sum of two vectors is perpendicular to their difference, it means that
the diagonals of the parallelogram are perpendicular.


But if a parallelogram has perpendicular diagonals, then this parallelogram is a rhombus,
and, hence, all his sides have equal lengths.


It is a geometric proof of your statement.


At this point, this geometric proof is complete.