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The problem asks for the modulus of the resultant of a vector system, given that the resultant is at its maximum. The vectors provided are at specific angles and have different magnitudes.

We are provided with the following vectors:

- A vector of magnitude \( 2u \) at an angle of \( 21^\circ \) above the X-axis (counter-clockwise).
- A vector of magnitude \( 8u \) along the Y-axis.
- A vector of magnitude \( \frac{2\sqrt{3}}{3} u \) at an angle of \( 9^\circ \) below the X-axis.

### To find the maximum resultant:
The resultant vector is at its maximum when the vectors are added head-to-tail in such a way that the resultant magnitude is the greatest. This happens when the vectors are aligned in the same direction or nearly the same direction.

### Solution Approach:
To solve this problem, the component method can be used. We will break each vector into its X and Y components and sum them up.

1. **Vector 1**: \( 2u \) at \( 21^\circ \)
   - \( x_1 = 2u \cos(21^\circ) \)
   - \( y_1 = 2u \sin(21^\circ) \)

2. **Vector 2**: \( 8u \) along the Y-axis
   - \( x_2 = 0 \) (no X component)
   - \( y_2 = 8u \) (since it is entirely in the Y direction)

3. **Vector 3**: \( \frac{2\sqrt{3}}{3} u \) at \( 9^\circ \)
   - \( x_3 = \frac{2\sqrt{3}}{3} u \cos(9^\circ) \)
   - \( y_3 = -\frac{2\sqrt{3}}{3} u \sin(9^\circ) \) (negative since it is below the X-axis)

The total X and Y components of the resultant vector can be calculated as:
\[
R_x = x_1 + x_2 + x_3
\]
\[
R_y = y_1 + y_2 + y_3
\]

Finally, the magnitude of the resultant vector \( R \) is given by:
\[
R = \sqrt{R_x^2 + R_y^2}
\]

I will now calculate the exact value of the modulus of the resultant.

Let me compute this.The modulus of the resultant vector is approximately \( 9.05u \). From the given options, the closest value is \( D) 10u \).

Would you like more details on any part of the solution?

Here are 5 related questions to expand your understanding:

1. How are vector components calculated for different angles?
2. Why is the resultant vector maximum when the vectors are closely aligned?
3. How do we calculate vector addition using components?
4. What happens if the vectors are perpendicular to each other?
5. Can you explain the significance of using trigonometric functions in vector analysis?

**Tip:** When breaking vectors into components, always ensure the angle is measured from the correct axis to avoid sign errors in the components.

A partir del sistema de vectores mostrados, determine el módulo de la resultante sabiendo que esta es máxima.

Learn how to determine the modulus of the resultant vector in a system with multiple vectors at various angles using trigonometry and vector addition. This problem involves calculating vector components and applying the Pythagorean theorem.

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