why these two vectors are orthogonal

Question

Let $E$ be a normed vector space. Moreover let $x \in E$ and suppose there are $y_1, y_2$ such that : $$\| x - y_1 \| = \|x - y_2 \|$$

Then my book say the following without any justifications (so I guess it should quite obvious ):

The vectors $(x-y_1) + (x-y_2)$ and $(x-y_1) - (x-y_2)$ are orthogonal

By taking the dot product is quite obvious that these two vectors are orthogonal, yet i don't see why it's geometrically true. I can't visualise it. I guess that if there aren' any justification it should geometrically obvious ?

Thank you !

Answer 1

Hint: it suffices to prove that the diagonals of a rhombus are perpendicular. This follows from a basic fact from high school geometry, namely that the vertex of an isoceles triangle lies on the perpendicular bisector of the base.

Answer 2

Taking the dot product and showing that they’re orthogonal is enough to show that it’s geometrically obvious. If you want to visualize it, just come up with some examples of $x$, $y_1$, and $y_2$ that satisfy your initial condition and draw it out.

Here’s one possible example.