The vectors $\mathbf{u}=\left(\genfrac{}{}{0ex}{}{4}{10}\right)$ and $\mathbf{v}=\left(\genfrac{}{}{0ex}{}{x}{x+3}\right)$ are parallel. Determine the value of $x$.

Let $\mathbf{a}$ and $\mathbf{b}$ be vectors such that $|\mathbf{a}|=3$, $|\mathbf{b}|=5$, and $|\mathbf{a}+\mathbf{b}|=7$. Find the cosine of the angle $\theta$ between vectors $\mathbf{a}$ and $\mathbf{b}$.

Determine the value of $k$ such that the vectors $\mathbf{a}=\left(\genfrac{}{}{0ex}{}{4}{k}\right)$ and $\mathbf{b}=\left(\genfrac{}{}{0ex}{}{-3}{6}\right)$ are perpendicular.

True or False: If two non-zero vectors $\mathbf{a}$ and $\mathbf{b}$ satisfy the condition that $\mathbf{a}+\mathbf{b}$ is perpendicular to $\mathbf{a}-\mathbf{b}$, then it must be true that $|\mathbf{a}|=|\mathbf{b}|$.

Two non-zero vectors $\mathbf{a}$ and $\mathbf{b}$ satisfy the relationship $\mathbf{a}\cdot \mathbf{b}=|\mathbf{a}||\mathbf{b}|$. What does this imply about the geometric relationship between the two vectors?