1. **Problem:** Use the parallelogram method to add vectors $\vec{x} = \langle 3, 4 \rangle$ and $\vec{y} = \langle 4, 1 \rangle$. 2. **Formula:** Vector addition by the parallelogram method is given by adding corresponding components: $$\vec{x} + \vec{y} = \langle x_1 + y_1, x_2 + y_2 \rangle$$ 3. **Calculation:** $$\vec{x} + \vec{y} = \langle 3 + 4, 4 + 1 \rangle = \langle 7, 5 \rangle$$ 4. **Explanation:** The parallelogram method involves placing the two vectors so their tails coincide, then completing the parallelogram. The diagonal from the common tail to the opposite corner represents the sum vector. 5. **Final answer:** $$\boxed{\langle 7, 5 \rangle}$$ This is the resultant vector from adding $\vec{x}$ and $\vec{y}$ using the parallelogram method.