1. **State the problem:** Find a vector of magnitude 7 in the direction of $ \mathbf{v} = 12\mathbf{i} - 5\mathbf{k} $. 2. **Formula and rules:** To find a vector of a given magnitude $ m $ in the direction of a vector $ \mathbf{v} $, use the formula: $$ \mathbf{u} = m \cdot \frac{\mathbf{v}}{\|\mathbf{v}\|} $$ where $ \|\mathbf{v}\| $ is the magnitude (length) of $ \mathbf{v} $. 3. **Calculate the magnitude of $ \mathbf{v} $:** $$ \|\mathbf{v}\| = \sqrt{12^2 + 0^2 + (-5)^2} = \sqrt{144 + 0 + 25} = \sqrt{169} = 13 $$ 4. **Find the unit vector in the direction of $ \mathbf{v} $:** $$ \frac{\mathbf{v}}{\|\mathbf{v}\|} = \frac{12\mathbf{i} - 5\mathbf{k}}{13} = \frac{12}{13}\mathbf{i} - \frac{5}{13}\mathbf{k} $$ 5. **Multiply the unit vector by the desired magnitude 7:** $$ \mathbf{u} = 7 \cdot \left( \frac{12}{13}\mathbf{i} - \frac{5}{13}\mathbf{k} \right) = \frac{7 \times 12}{13}\mathbf{i} - \frac{7 \times 5}{13}\mathbf{k} = \frac{84}{13}\mathbf{i} - \frac{35}{13}\mathbf{k} $$ 6. **Final answer:** $$ \boxed{\mathbf{u} = \frac{84}{13}\mathbf{i} - \frac{35}{13}\mathbf{k}} $$ This vector has magnitude 7 and points in the same direction as $ \mathbf{v} $.