Orthogonality of dot product is applicable only in $\mathbb{R}^3$ and $\mathbb{R}^7$

Question

Does orthogonality of the dot product (i.e., If the dot product of two vectors is $0$ then they are orthogonal to each other) apply only in $\mathbb{R}^3$ and $\mathbb{R}^7$? The reason is that I also have come across videos using dot product, in the context of orthogonality, in $\mathbb{R}^4$ and $\mathbb{R}^n$.

Answer 1

Often the definition of the angle between two vectors in $\mathbb{R}^n$ is based on the inverse cosine of their dot-product. That is, vectors are defined as being orthogonal if their dot product is zero.