To find the terminal point of the vector $\mathbf{u} = (3, -1)$ with an initial point $(2, 4)$:

Step-by-step Solution:

1. Recall the formula for a terminal point: If the initial point of a vector is $(x_1, y_1)$ and the vector is $\mathbf{u} = (a, b)$, the terminal point $(x_2, y_2)$ is calculated as: $x_2 = x_1 + a, \quad y_2 = y_1 + b$

2. Substitute the values:

   • Initial point: $(x_1, y_1) = (2, 4)$
   • Vector components: $\mathbf{u} = (3, -1)$

   Using the formula: $x_2 = 2 + 3 = 5$ $y_2 = 4 + (-1) = 3$

3. Final Answer: The terminal point is $(5, 3)$.

Would you like a detailed visualization of this solution or additional clarification? Here are related questions for deeper understanding:

1. What is the difference between the initial and terminal points of a vector?
2. How would the terminal point change if the vector components were negative?
3. Can you sketch the vector $\mathbf{u}$ with its initial point and terminal point on a Cartesian plane?
4. How does the length (magnitude) of the vector relate to its components?
5. How would the calculation change if the initial point were shifted to a different location?

Tip: To confirm your results graphically, sketch the initial point and add the vector components step-by-step on the Cartesian plane!