When is an inner product on $\mathbb{R}^n$ positive for non-equal vectors?

Question

Let $u_1, u_2, u_3 \in \mathbb{R}^n$ and $M \in \mathbb{R}^{n \times n}$. Let $\left< \cdot, \cdot \right>$ denote an inner product on $\mathbb{R}^n$.

Under what conditions on these objects and on $\left< \cdot, \cdot \right>$ do we have

$$\left< Mu_1 + u_2, \; u_3 - u_1 \right> > 0,$$

where $$Mu_1 + u_2 \neq u_3 - u_1 \ ?$$

More generally, when is an inner product on $\mathbb{R}^n$ positive for non-equal vectors? I'm surprised at how difficult it is to search an answer to this seemingly basic question.

Thank you for the help.

Answer 1

Let $a$ and $b$ be two linearly independent vectors. If the smaller angle between $a$ and $b$ is within $]0,\pi/2]$ the inner product will be positive, otherwise negative.