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The problem presents three vectors in a Cartesian coordinate system with their magnitudes. The goal is to calculate the magnitude of the resultant vector.

We have the following information:

1. One vector with a magnitude of \( 4 \) in the negative \( x \)-axis direction.
2. One vector with a magnitude of \( 3 \) in the negative \( y \)-axis direction.
3. A vector with a magnitude of \( 10 \), making an angle of \( 37^\circ \) with the positive \( x \)-axis.

### Solution Strategy:
- Decompose the third vector into its components: \( x \)-component and \( y \)-component.
- Sum the components of the vectors along the \( x \)-axis and \( y \)-axis.
- Use the Pythagorean theorem to calculate the magnitude of the resultant vector.

Let's start by decomposing the vector with magnitude \( 10 \) and an angle of \( 37^\circ \).

- The \( x \)-component of the vector is \( 10 \cos(37^\circ) \).
- The \( y \)-component of the vector is \( 10 \sin(37^\circ) \).

Now, let's calculate the components and the resultant vector.Here are the results:

- The \( x \)-component of the vector with magnitude \( 10 \) is approximately \( 7.99 \).
- The \( y \)-component of the vector with magnitude \( 10 \) is approximately \( 6.02 \).

Now, summing all vectors:

- The total \( x \)-component of the resultant vector is \( 3.99 \).
- The total \( y \)-component of the resultant vector is \( 3.02 \).

Finally, the magnitude of the resultant vector is approximately **5.00**.

Would you like further details on the calculations, or have any questions?

Here are 5 related questions you might want to explore:
1. How do we handle vector addition geometrically?
2. What are the other methods to decompose vectors into components?
3. How would the problem change if the angle were different?
4. Can vectors in 3D space be calculated similarly?
5. How does changing the magnitude of one vector affect the resultant?

**Tip:** Always decompose vectors into components when adding them in different directions; this simplifies the calculation significantly.

En el esquema se muestran tres vectores ubicados en un sistema de coordenadas cartesianas con sus respectivos módulos. Calcule el módulo del vector resultante.

This example demonstrates how to calculate the magnitude of a resultant vector from three given vectors in Cartesian coordinates. The process involves vector decomposition using trigonometric functions and applying the Pythagorean theorem.

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