Solution:

1. Find the vector 𝘂 in component form: A vector in component form is expressed as: $\mathbf{u} = \langle x_2 - x_1, y_2 - y_1 \rangle$ Here, the initial point is $(x_1, y_1) = (2, 1)$ and the terminal point is $(x_2, y_2) = (3, 2)$.

   Subtract the coordinates: $\mathbf{u} = \langle 3 - 2, 2 - 1 \rangle = \langle 1, 1 \rangle$

   So: $\mathbf{u} = \langle 1, 1 \rangle$

2. Vector 𝘃 is not given, but assuming it has coordinates (𝑣ₓ, 𝑣ᵧ): $\mathbf{v} = \langle v_x, v_y \rangle$

3. Find $2\mathbf{u}$: Multiply each component of $\mathbf{u}$ by 2: $2\mathbf{u} = 2 \cdot \langle 1, 1 \rangle = \langle 2, 2 \rangle$

4. Find $\mathbf{u} + \mathbf{v}$: Add the corresponding components of $\mathbf{u}$ and $\mathbf{v}$: $\mathbf{u} + \mathbf{v} = \langle 1 + v_x, 1 + v_y \rangle$

   If the values of $v_x$ and $v_y$ are specified, substitute them into this formula to get the final result.

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Final Answers:

1. 𝘂 = $\langle 1, 1 \rangle$
2. 2𝘂 = $\langle 2, 2 \rangle$
3. 𝘂 + 𝘃 = $\langle 1 + v_x, 1 + v_y \rangle$

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Would you like more details or have any specific values for $\mathbf{v}$? Let me know!

Related Questions:

1. How do you determine the magnitude of a vector given its components?
2. What is the unit vector of $\mathbf{u} = \langle 1, 1 \rangle$?
3. How do you find the dot product of two vectors?
4. What is the geometric interpretation of vector addition and scalar multiplication?
5. How do you determine the angle between two vectors using their components?

Tip: Always represent vectors in their component form for easy calculations and visualization in coordinate geometry.