The problem requires finding a point symmetrical to $P(1, 0, -1)$ relative to the given line:

$\frac{x+1}{1} = \frac{y-1}{2} = \frac{z}{3}.$

Let's solve this step by step:

1. Parametric Equation of the Line

The symmetric point is relative to the given line. To describe the line parametrically:

• Point on the line: $(-1, 1, 0)$
• Direction vector: $(1, 2, 3)$

The parametric equations are: $x = -1 + t, \quad y = 1 + 2t, \quad z = 3t,$ where $t$ is the parameter.

2. Find the Projection of $P$ on the Line

To find the symmetric point, we first need the projection of $P(1, 0, -1)$ onto the line. This involves finding $t$ such that the vector from $P$ to a point on the line is perpendicular to the line's direction vector.

• A point on the line: $(-1 + t, 1 + 2t, 3t)$
• Vector from $P$ to this point: $\vec{PQ} = (-1+t-1, 1+2t-0, 3t+1) = (t-2, 1+2t, 3t+1).$ This vector is perpendicular to the direction vector $(1, 2, 3)$ if their dot product is zero: $(t-2)(1) + (1+2t)(2) + (3t+1)(3) = 0.$ Simplify: $t - 2 + 2 + 4t + 9t + 3 = 0 \implies 14t + 3 = 0 \implies t = -\frac{3}{14}.$

3. Coordinates of the Projection Point

Substitute $t = -\frac{3}{14}$ into the parametric equations: $x = -1 + t = -1 - \frac{3}{14} = -\frac{17}{14},$ $y = 1 + 2t = 1 + 2\left(-\frac{3}{14}\right) = 1 - \frac{6}{14} = \frac{8}{14} = \frac{4}{7},$ $z = 3t = 3\left(-\frac{3}{14}\right) = -\frac{9}{14}.$ The projection point is: $Q\left(-\frac{17}{14}, \frac{4}{7}, -\frac{9}{14}\right).$

4. Symmetrical Point Calculation

The symmetrical point $P'$ is found using: $P' = 2Q - P.$ Substitute the coordinates of $Q$ and $P(1, 0, -1)$: $x' = 2\left(-\frac{17}{14}\right) - 1 = -\frac{34}{14} - \frac{14}{14} = -\frac{48}{14} = -\frac{24}{7},$ $y' = 2\left(\frac{4}{7}\right) - 0 = \frac{8}{7},$ $z' = 2\left(-\frac{9}{14}\right) - (-1) = -\frac{18}{14} + \frac{14}{14} = -\frac{4}{14} = -\frac{2}{7}.$ The symmetrical point is: $P'\left(-\frac{24}{7}, \frac{8}{7}, -\frac{2}{7}\right).$

Would you like further clarification or details? 😊

Related Questions:

1. How do you calculate the parametric equations of a line in 3D?
2. What is the mathematical meaning of the dot product in geometry?
3. How can you verify if a point lies on a given line in 3D space?
4. What are alternative methods for finding a symmetric point relative to a line?
5. How is the projection of a point onto a line useful in real-world applications?

Tip:

Always check your calculations by substituting the final point back into the geometric constraints to confirm symmetry or alignment!