1. **State the problem:** Compute the norm (magnitude) of the vector $\mathbf{v} = [42, 14]$ using properties of the dot product and norm. 2. **Recall the formula:** The norm of a vector $\mathbf{v} = [v_1, v_2]$ is given by $$\|\mathbf{v}\| = \sqrt{\mathbf{v} \cdot \mathbf{v}} = \sqrt{v_1^2 + v_2^2}$$ This comes from the dot product definition $\mathbf{v} \cdot \mathbf{v} = v_1^2 + v_2^2$. 3. **Apply the formula:** For $\mathbf{v} = [42, 14]$, compute $$\|\mathbf{v}\| = \sqrt{42^2 + 14^2}$$ 4. **Calculate the squares:** $$42^2 = 1764$$ $$14^2 = 196$$ 5. **Sum the squares:** $$1764 + 196 = 1960$$ 6. **Take the square root:** $$\|\mathbf{v}\| = \sqrt{1960}$$ 7. **Simplify the square root:** Note that $1960 = 196 \times 10$ and $\sqrt{196} = 14$, so $$\|\mathbf{v}\| = 14 \sqrt{10}$$ **Final answer:** $$\boxed{14 \sqrt{10}}$$ This is the magnitude of the vector $[42, 14]$.