Cauchy-Schwarz, the Triangle Inequality, Neighborhoods and Paths

Cauchy-Schwarz follows from a quadratic that cannot cross zero

The triangle inequality is Cauchy-Schwarz applied once

Neighborhoods describe closeness with open balls

Paths connect points and define connected sets

Vectors $\mathbf{a}$ and $\mathbf{b}$ satisfy $\|\mathbf{a}\| = 3$ and $\|\mathbf{b}\| = 5$. What is the largest value that $|\mathbf{a} \cdot \mathbf{b}|$ can take, and when is it achieved?

Let $\mathbf{a} = (1, 2)$ and $\mathbf{b} = (3, -1)$. What is the exact length $\|\mathbf{a} + \mathbf{b}\|$?

Consider the open ball $B(\mathbf{0}, 2)$ in the plane, all points at distance strictly less than $2$ from the origin. Which point lies inside it?

Which parametrized path travels from $\mathbf{p}$ to $\mathbf{q}$ along a straight line as $t$ runs from $0$ to $1$?

For $\mathbf{a} = (1, 2)$ and $\mathbf{b} = (3, -1)$, give the value of the right side $\|\mathbf{a}\|\,\|\mathbf{b}\|$ of the Cauchy-Schwarz inequality, in the form a*sqrt(b) with integers, like 2*sqrt(3).

Give the distance from the origin to the point $(3, 4)$, which is the threshold radius that any open ball centered at the origin must exceed to contain the point. Give the exact value as a number.