To find the arc length parameterization of the line given by:

$x = 7 + 7t, \quad y = 3 - 4t, \quad z = 4 + 4t$

we start by finding the direction vector of the line. The direction vector $\mathbf{d}$ can be extracted from the coefficients of $t$:

$\mathbf{d} = \langle 7, -4, 4 \rangle$

Next, we calculate the magnitude of this direction vector, $|\mathbf{d}|$:

$|\mathbf{d}| = \sqrt{7^2 + (-4)^2 + 4^2} = \sqrt{49 + 16 + 16} = \sqrt{81} = 9$

The arc length $s$ is related to the parameter $t$ by:

$s = |\mathbf{d}| \cdot t = 9t$ $t = \frac{s}{9}$

Now, substituting $t = \frac{s}{9}$ into the original parametric equations gives us the arc length parameterization:

$x(s) = 7 + 7\left(\frac{s}{9}\right) = 7 + \frac{7s}{9}$ $y(s) = 3 - 4\left(\frac{s}{9}\right) = 3 - \frac{4s}{9}$ $z(s) = 4 + 4\left(\frac{s}{9}\right) = 4 + \frac{4s}{9}$

Therefore, the arc length parameterization is:

$x(s) = 7 + \frac{7s}{9}$ $y(s) = 3 - \frac{4s}{9}$ $z(s) = 4 + \frac{4s}{9}$

Now, to find the point on the line that is 11 units from the reference point $(7, 3, 4)$ in the direction of increasing parameter $s$, we set $s = 11$:

$x(11) = 7 + \frac{7 \cdot 11}{9} = 7 + \frac{77}{9} = 7 + 8.555... = 15.555...$ $y(11) = 3 - \frac{4 \cdot 11}{9} = 3 - \frac{44}{9} = 3 - 4.888... = -1.888...$ $z(11) = 4 + \frac{4 \cdot 11}{9} = 4 + \frac{44}{9} = 4 + 4.888... = 8.888...$

Therefore, the point on the line that is 11 units from the reference point in the direction of increasing parameter is approximately:

$\left(15.56, -1.89, 8.89\right)$

Would you like more details or have any questions? Here are some related questions you might want to ask:

1. How do you find the direction vector of a line?
2. What is the significance of arc length parametrization in vector calculus?
3. How do you calculate the magnitude of a vector?
4. How do you convert between different parameterizations of a line?
5. How can you verify that a point lies on a given line?

Tip: Always check your calculations carefully when dealing with vectors and their magnitudes to ensure accuracy in your results.