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All of the theorems I have found in my textbook and on the internet just state the converse; is it true both ways?

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2 Answers

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Here is a way to see it in $\mathbb R^n$.

Let $a,b \neq 0$ two vectors. Since $\langle a, b \rangle = \|a\| \cdot \|b\| \cdot \cos(\phi)$ where $\phi$ is the angle between $a$ and $b$, you get $\langle a, b \rangle = 0 \iff \phi = ±π/2 \iff a$ and $b$ are orthogonal.

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Yes since the dot product of two NON ZERO vectors is the product of the norm (length) of each vector and cosine the angle between them. If the dot product is zero then the cosine is zero then the angle between the 2 vectors is 90 degress hence orthogonal

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