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We need to find the magnitude of the x-component of the resultant vector \(\vec{R}\), which is the sum of vectors \(\vec{A}\) and \(\vec{B}\):

\[
\vec{R} = \vec{A} + \vec{B}
\]

### Step 1: Break down the vectors into components

- \(\vec{A} = 12.0 \, \text{m}\) directed along the negative x-axis.
  So, the components of \(\vec{A}\) are:
  \[
  A_x = -12.0 \, \text{m}, \quad A_y = 0
  \]

- \(\vec{B} = 18.0 \, \text{m}\) at an angle of \(37.0^\circ\) above the positive x-axis.
  The components of \(\vec{B}\) are:
  \[
  B_x = 18.0 \cos(37.0^\circ), \quad B_y = 18.0 \sin(37.0^\circ)
  \]

### Step 2: Find the x-component of \(\vec{R}\)

The x-component of \(\vec{R}\) is the sum of the x-components of \(\vec{A}\) and \(\vec{B}\):
\[
R_x = A_x + B_x = -12.0 + 18.0 \cos(37.0^\circ)
\]

Let's calculate that.The x-component of the resultant vector \(\vec{R}\) is approximately \(2.38 \, \text{m}\).

Do you have any questions or would you like further details? 

Here are some related questions to explore:
1. How would you find the magnitude of the resultant vector \(\vec{R}\)?
2. What are the y-components of vectors \(\vec{A}\) and \(\vec{B}\)?
3. How does the angle of vector \(\vec{B}\) affect its components?
4. What would happen to \(\vec{R}\) if vector \(\vec{A}\) was directed along the positive x-axis?
5. How could you use vector addition to solve problems involving more than two vectors?

**Tip:** Always resolve vectors into their components along the coordinate axes when solving vector addition problems!

When vector A is added to vector B, the resultant is vector R such that R = A + B. What is the magnitude of the x-component of the resultant vector R?

Learn how to find the magnitude of the x-component of a resultant vector using vector addition. This problem involves breaking vectors into components and applying trigonometry to calculate the x-component. Suitable for students in Grades 9-12.

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