Vectors

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Hi, I'm learning some basic linear algebra and have a question.

If there are two vectors (let's say v and w) that have the same length (not 0) but different direction, then the resultant vectors (v+w and v-w) always seem to be perpendicular to each other. That is, the dot product of v+w and v-w seem to be always 0.

I made this sketch to illustrate. I'm curious to know exactly why does this happen?