https://r2.erweima.ai/mathbot/image/17266417434359x25cmbf.jpg

In the provided image, there are three vectors positioned on a Cartesian coordinate system, and we are asked to calculate the magnitude of the resultant vector. Let's break down the steps.

### Given Data:
1. **Vector 1**: Magnitude 4 along the negative x-axis.
2. **Vector 2**: Magnitude 3 along the negative y-axis.
3. **Vector 3**: Magnitude 10 at an angle of \(37^\circ\) from the positive x-axis.

### Steps to solve:

1. **Vector Decomposition**:
   - **Vector 1**: This vector lies along the x-axis, so its components are:
     \[
     \mathbf{v_1} = (-4, 0)
     \]
   - **Vector 2**: This vector lies along the y-axis, so its components are:
     \[
     \mathbf{v_2} = (0, -3)
     \]
   - **Vector 3**: Since this vector is at an angle of \(37^\circ\) from the positive x-axis, we decompose it into x and y components:
     \[
     \mathbf{v_3x} = 10 \cdot \cos(37^\circ)
     \]
     \[
     \mathbf{v_3y} = 10 \cdot \sin(37^\circ)
     \]

2. **Calculate Components**:
   Using the approximate trigonometric values:
   \[
   \cos(37^\circ) \approx 0.7986, \quad \sin(37^\circ) \approx 0.6018
   \]
   So, the components of **Vector 3** are:
   \[
   \mathbf{v_3x} = 10 \cdot 0.7986 = 7.986
   \]
   \[
   \mathbf{v_3y} = 10 \cdot 0.6018 = 6.018
   \]
   Hence, the vector is:
   \[
   \mathbf{v_3} = (7.986, 6.018)
   \]

3. **Sum of Components**:
   Now, add the components of the three vectors:
   - x-components:
     \[
     x_{total} = -4 + 0 + 7.986 = 3.986
     \]
   - y-components:
     \[
     y_{total} = 0 - 3 + 6.018 = 3.018
     \]

4. **Magnitude of the Resultant Vector**:
   The magnitude \(R\) of the resultant vector is given by the Pythagorean theorem:
   \[
   R = \sqrt{x_{total}^2 + y_{total}^2}
   \]
   Substituting the values:
   \[
   R = \sqrt{(3.986)^2 + (3.018)^2}
   \]
   Let's calculate the final result.The magnitude of the resultant vector is approximately \( 5.00 \) units.

Would you like more details or have any other questions? Here are some related questions for further exploration:

1. How do you calculate the direction of the resultant vector?
2. What happens if the angle of the third vector changes to 45°?
3. Can you find the resultant vector if all vectors were in three dimensions?
4. How do vector components relate to the physical concept of forces?
5. How do you determine the equilibrium of forces acting on a point?

**Tip:** Always break vectors into components for easier calculation of the resultant when working with non-parallel vectors.

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

Learn how to calculate the magnitude of a resultant vector in a Cartesian plane from three vectors with given magnitudes. The problem involves vector decomposition, trigonometry, and the Pythagorean theorem. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation