Introduction to 3D Geometry and Perpendicular Lines

In three-dimensional (3D) space, understanding the geometry and algebra involved in finding the equation of a line that is perpendicular to another line from a given point, is a fundamental skill in mathematics. This article will explore the process of finding such a line using the given problem: the line x5/1 y3/4 z-6/-9, and the point (2, -1) where the perpendicular is drawn from.

Understanding the Given Line and Point

The given line is in parametric form:

x5/1 y3/4 z-6/-9 R

This means that each coordinate of the line can be expressed in terms of a single parameter R:

x R - 5 y 4R - 3 z -9R - 6

The point from which the perpendicular is drawn is (2, -1). This point, denoted as A(2, -1), will be used to find the equation of the line that is perpendicular to the given line.

Deriving the Direction Ratio of the Perpendicular Line

The direction ratios of line AB, where B is the point of intersection on the given line, can be found using the following relation for perpendicularity:

(p - 2)(4) (q - 4)(-9) (r 1)(-27) 0

Simplifying, we get:

4p - 9q - 54r - 27 0 … Equation 1

From the given line, we have the direction ratios:

p R - 5 q 4R - 3 r -9R - 6

Substituting these into Equation 1, we solve for R:

(R - 5)4 (4R - 3)(-9) (-9R - 6)(-27) 0

4R - 20 - 36R 27 - 243R - 162 0

-275R - 155 0

R 1

Thus, the direction ratios of the line are:

p - 2 -1 q - 4 -3 r 1 -7

Evaluating the Perpendicular Line Equation

The equation of the perpendicular line AB is then given by:

(x - 2) / 2 (y - 4) / 4 (z 1) / -7

Simplifying, we get:

x - 2 -1 / 2 * (x - 2) / 2 y - 4 -3 / 4 * (y - 4) / 4 z 1 -7 / -7 * (z 1) / -7

Simplifying further, we have:

x - 2 -1 / 2 y - 4 -3 / 4 z 1 -1 / 7

Thus, the equation of the perpendicular line is:

x - 2 / 6 y - 4 / 3 z 1 / 2

Conclusion

In conclusion, the process of finding the equation of a line that is perpendicular to another line from a given point involves vector algebra and parametric equations. The key steps involve expressing the given line in parametric form, using the condition for perpendicularity to derive the direction ratios, and substituting these into the general form of the line equation.