The problem asks for the direction of the negative of the given vector, which has components $(24.1, 58.8)$, with the direction measured counterclockwise from the positive $x$-axis.

Step 1: Find the direction of the original vector

To find the angle $\theta_{\text{original}}$ for the original vector $(24.1, 58.8)$:

$\tan(\theta_{\text{original}}) = \frac{y}{x} = \frac{58.8}{24.1}$ First, let's compute this tangent value:

$\tan(\theta_{\text{original}}) = \frac{58.8}{24.1} \approx 2.44$

Now, take the arctangent of this value to find the angle:

$\theta_{\text{original}} = \tan^{-1}(2.44)$

Let’s calculate that.

$\theta_{\text{original}} \approx 67.6^\circ$

Thus, the direction of the original vector is approximately $67.6^\circ$ counterclockwise from the positive $x$-axis.

Step 2: Find the direction of the negative of the vector

The negative of a vector $(-x, -y)$ points in the opposite direction of the original vector. To find the angle of the negative vector, we add $180^\circ$ to the original angle:

$\theta_{\text{negative}} = \theta_{\text{original}} + 180^\circ = 67.6^\circ + 180^\circ = 247.6^\circ$

Thus, the angle of the negative vector is approximately 247.6°.

Final Answer:

The angle of the negative of the vector $(24.1, 58.8)$, measured counterclockwise from the positive $x$-axis, is 247.6°.

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Do you want more details or have any other questions?

Here are some related questions you might consider:

1. How would the angle change if one or both components were negative?
2. How do you calculate the magnitude of this vector?
3. What would the angle be if the vector components were swapped?
4. How is the direction of a vector affected if it is multiplied by a scalar?
5. What if the vector was measured in a different coordinate system?

Tip: Always visualize vectors on the coordinate plane to better understand their direction and magnitude!