To find the Cartesian equation of the line passing through the point $(-2, 4, -5)$ and parallel to the given line, we follow these steps:

Step 1: Identify the direction vector of the given line

The given line is described by the equation:

$\frac{x + 3}{3} = \frac{y - 4}{5} = \frac{z + 8}{6}$

This is a symmetric form equation. We can break it down to identify the direction vector of the line. The equation can be written as:

$\frac{x + 3}{3} = \frac{y - 4}{5} = \frac{z + 8}{6} = t$

This leads to parametric equations:

$x = 3t - 3, \quad y = 5t + 4, \quad z = 6t - 8$

The direction vector of this line is $\vec{d} = \langle 3, 5, 6 \rangle$, which represents the coefficients of $t$ in the parametric equations.

Step 2: Equation of the new line

The new line is parallel to the given line, so it will have the same direction vector $\vec{d} = \langle 3, 5, 6 \rangle$, and it passes through the point $(-2, 4, -5)$.

We can use the parametric form of the line:

$x = -2 + 3t, \quad y = 4 + 5t, \quad z = -5 + 6t$

Step 3: Convert to Cartesian form

To convert the parametric equations to Cartesian form, eliminate $t$ from each equation. Start with:

$x + 2 = 3t, \quad y - 4 = 5t, \quad z + 5 = 6t$

Solve each equation for $t$:

$t = \frac{x + 2}{3}, \quad t = \frac{y - 4}{5}, \quad t = \frac{z + 5}{6}$

Now, equate the expressions for $t$:

$\frac{x + 2}{3} = \frac{y - 4}{5} = \frac{z + 5}{6}$

This is the Cartesian equation of the line passing through $(-2, 4, -5)$ and parallel to the given line.

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Related Questions:

1. How would you find the vector equation of a line passing through two points?
2. What does the direction vector of a line represent in 3D geometry?
3. How can you convert a parametric equation of a line into symmetric form?
4. How can the concept of parallel lines be applied in 3D space?
5. What is the relationship between the direction ratios and direction cosines of a line?

Tip:

When working with lines in 3D, remember that the direction vector provides the direction of movement along the line, and the parametric equations are the key to describing the line fully.