https://file.aiquickdraw.com/mathbot/file/1726308826843qv9zs7v0.jpg

To find the distance from a point to a line in 3D space, we need to use the formula for the distance between a point \((x_1, y_1, z_1)\) and a line given in parametric form \(\mathbf{r}(t) = \mathbf{r}_0 + t\mathbf{v}\), where:

- \(\mathbf{r}_0\) is a point on the line,
- \(\mathbf{v}\) is the direction vector of the line,
- and \(\mathbf{P} = (x_1, y_1, z_1)\) is the point we are measuring the distance from.

The formula for the shortest distance \(d\) between a point \(\mathbf{P}(x_1, y_1, z_1)\) and a line is:

\[
d = \frac{|\mathbf{v} \times (\mathbf{P} - \mathbf{r}_0)|}{|\mathbf{v}|}
\]

From the image, the parametric equations of the line are:

\[
x = 6 + t, \quad y = 2t, \quad z = 24\sqrt{5} + 2t
\]

This gives:
- \(\mathbf{r}_0 = (6, 0, 24\sqrt{5})\) as a point on the line when \(t = 0\),
- \(\mathbf{v} = (1, 2, 2)\) as the direction vector of the line,
- \(\mathbf{P} = (6, 0, 0)\) as the point from which we need to find the distance.

Now, let's calculate the distance using the formula above.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you want to walk through the calculations manually or have additional questions, feel free to ask!

Here are some related questions for practice:
1. How do you find the direction vector of a parametric line?
2. What is the formula for the cross product of two vectors in 3D?
3. How does the formula for distance from a point to a line in 3D differ from 2D?
4. How can you find the parametric form of a line from two points in 3D?
5. What is the geometric interpretation of the cross product in the context of lines and points?

**Tip:** When finding the shortest distance from a point to a line, always remember that the direction vector of the line plays a key role in the calculation!

Find the distance from the point (6, 0, 0) to the line x = 6 + t, y = 2t, z = 24√5 + 2t.

This problem covers how to calculate the distance from a point (6, 0, 0) to a line described by parametric equations. Using vector cross products and 3D geometry concepts, learn to solve for the shortest distance. Suitable for advanced high school or undergraduate students.

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