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!