What is the Distance of a Point from a Plane Measured Parallel to a Line?

Today, we will explore a mathematical problem involving the determination of the distance between a point and a plane, measured parallel to a specific line. This is a common task in geometry and can be particularly useful in fields such as computer graphics, engineering, and physics.

Problem Statement

We are given the following:

Point: (10, -3, -3) Plane: x - y - z 9 Line: (frac{x-2}{2} frac{y-3}{3} frac{z-6}{-6})

The goal is to find the distance from the point to the plane, measured along a line parallel to the given line.

Solution

First, let's identify the direction ratios (DR) of the given line:

2, 3, -6

The parametric equation of the line passing through the point (10, -3, -3) and parallel to the given line can be written as:

(frac{x-10}{2} frac{y 3}{3} frac{z 3}{-6} lambda)

Let's assume a general point on this line as:

x 10 2lambda y -3 3lambda z -3 - 6lambda

This point lies on the given plane, so it must satisfy the plane equation:

((10 2lambda) - (-3 3lambda) - (-3 - 6lambda) 9)

Simplifying, we get:

10 2lambda 3 - 3lambda 3 6lambda 9)

2lambda 6lambda - 3lambda 9 - 16)

5lambda 5)

(lambda 1)

Substituting (lambda 1) into the parametric equations, we find the point on the plane:

x 10 2(1) 12)

y -3 3(1) 0)

z -3 - 6(1) -9)

Therefore, the point of intersection is (12, 0, -9).

The distance between (10, -3, -3) and (12, 0, -9) is given by:

(k sqrt{(12-10)^2 (0 3)^2 (-9 3)^2})

k sqrt{2^2 3^2 6^2})

k sqrt{4 9 36})

k sqrt{49})

k 7)

Step-by-Step Solution

Identify the direction ratios: 2, 3, -6. Parametric equations of the line: (frac{x-10}{2} frac{y 3}{3} frac{z 3}{-6} lambda). General point on the line: (10 2(lambda), -3 3(lambda), -3 - 6(lambda)). Substitute into the plane equation: ((10 2lambda) - (-3 3lambda) - (-3 - 6lambda) 9). Solve for (lambda): (lambda 1). Substitute (lambda 1) into the parametric equations to find the point (12, 0, -9). Calculate the distance: (k sqrt{49}), which equals 7.

Conclusion

We have successfully determined the distance from the point (10, -3, -3) to the plane x - y - z 9, measured parallel to the line (frac{x-2}{2} frac{y-3}{3} frac{z-6}{-6}). This process involves identifying the parametric equations of the line, finding the point of intersection with the plane, and then calculating the distance between the point and the intersection point.

Understanding how to solve such problems is essential in various applications where we need to measure distances in specific directions. This knowledge can be applied in fields like computer graphics, engineering, and physics.