1. **Problem statement:** Find the magnitudes (modules) of vectors $\overline{AD}$, $\overline{AC}$, and $\overline{AC_1}$ in a cube $ABCDA_1B_1C_1D_1$ with edge length 1. 2. **Formula and rules:** The magnitude of a vector between two points is the length of the segment connecting them. For right triangles, use the Pythagorean theorem: $$c = \sqrt{a^2 + b^2}$$ where $c$ is the hypotenuse. 3. **Step 1: Find $|\overline{AD}|$** Since $AD$ is an edge of the cube with length 1, $$|\overline{AD}| = 1$$ 4. **Step 2: Find $|\overline{AC}|$** Triangle $\triangle ADC$ is right-angled at $D$, with legs $AD = 1$ and $DC = 1$. Using Pythagoras: $$AC^2 = AD^2 + DC^2 = 1^2 + 1^2 = 2$$ So, $$|\overline{AC}| = AC = \sqrt{2}$$ 5. **Step 3: Find $|\overline{AC_1}|$** Triangle $\triangle ACC_1$ is right-angled at $C$, with legs $AC = \sqrt{2}$ and $CC_1 = 1$. Using Pythagoras: $$AC_1^2 = AC^2 + CC_1^2 = (\sqrt{2})^2 + 1^2 = 2 + 1 = 3$$ So, $$|\overline{AC_1}| = AC_1 = \sqrt{3}$$ **Final answer:** $$|\overline{AD}| = 1; \quad |\overline{AC}| = \sqrt{2}; \quad |\overline{AC_1}| = \sqrt{3}$$