To determine the vector \( \mathbf{v} \) given its initial point \( (1, 2) \) and terminal point \( (4, 4) \), we first express the vector in component form. The components of the vector can be calculated by subtracting the coordinates of the initial point from those of the terminal point. Specifically, the x-component \( v_x \) is calculated as:

\[v_x = x_2 - x_1 = 4 - 1 = 3\]

Similarly, the y-component \( v_y \) is calculated as:

\[v_y = y_2 - y_1 = 4 - 2 = 2\]

Thus, the vector \( \mathbf{v} \) in component form is \( \mathbf{v} = (3, 2) \).

Next, we can sketch the position vector, which starts at the origin \( (0, 0) \). To do this, we move 3 units along the x-axis and 2 units along the y-axis, resulting in the endpoint at \( (3, 2) \). This visual representation helps in understanding the direction and magnitude of the vector.

To find the magnitude of vector \( \mathbf{v} \), we use the formula for the magnitude of a vector:

\[\|\mathbf{v}\| = \sqrt{v_x^2 + v_y^2}\]

Substituting the values of \( v_x \) and \( v_y \) into the equation gives:

\[\|\mathbf{v}\| = \sqrt{3^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13}\]

Therefore, the magnitude of vector \( \mathbf{v} \) is \( \sqrt{13} \). This concludes the calculation of the vector's component form, its sketch as a position vector, and its magnitude.