1. **State the problem:** We are given a vector $\mathbf{v} = \begin{pmatrix} -2 \\ 7 \end{pmatrix}$ and need to find its modulus (magnitude) $|\mathbf{v}|$. 2. **Formula:** The modulus of a vector $\mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix}$ is given by: $$ |\mathbf{v}| = \sqrt{x^2 + y^2} $$ This formula comes from the Pythagorean theorem, treating the vector components as legs of a right triangle. 3. **Apply the formula:** Substitute $x = -2$ and $y = 7$: $$ |\mathbf{v}| = \sqrt{(-2)^2 + 7^2} = \sqrt{4 + 49} $$ 4. **Simplify:** $$ |\mathbf{v}| = \sqrt{53} $$ Since 53 is a prime number, it cannot be simplified further. 5. **Final answer:** The modulus of the vector $\mathbf{v}$ is: $$ |\mathbf{v}| = \sqrt{53} $$ This is the exact value expressed as a surd.