Introduction
The process of finding the projection of a line on a plane and determining the intersection of lines with planes is a fundamental concept in analytic geometry. This article delves into the detailed steps and calculations required to solve such problems, ensuring a clear and comprehensive understanding of the underlying principles.
Projection of a Line on a Plane
Consider the line defined by the parametric equations:
x1/-1 y/2 z-1/3 t
where x1, y, and z are coordinates along the line, and t is the parameter. The plane in question is defined by the equation:
x - 2y - z 6
Our goal is to find the projection of this line on the given plane. This involves determining the points of projection and the associated vector.
Step-by-Step Solution
1. Parametric Form of the Line
First, let's express the line in parametric form:
x1 -t, y 2t, z 3t 1
where t is the parameter.
2. Intersection Point on the Plane
Substitute the parametric equations of the line into the plane equation:
-t - 2(2t) - (3t 1) 6
Simplify the equation:
-t - 4t - 3t - 1 6
-8t - 1 6
-8t 7
t -7/8
Substituting this value back into the parametric equations gives the coordinates of the point of intersection:
x1 7/8, y -7/4, z 19/8
So, the point of intersection is A(7/8, -7/4, 19/8).
3. Projection Point on the Plane
Let's denote the projection of point A on the plane as B. We need to determine the parameter value for point B using the plane equation.
The parametric equations of the line can be written as:
x -t, y 2t k, z 3t 1 - m
To find the projection of point A on the plane, we set the coordinates of A in the plane equation:
-(-7/8) - 2(2(-7/8) - k) - (3(-7/8) 1 - m) 6
14/8 - 2(-14/8 - k) - (-21/8 1 - m) 6
14/8 28/8 2k - (-21/8 8/8 - m) 6
42/8 2k 21/8 - 8/8 m 6
63/8 2k m 6
2k m 6 - 63/8
2k m 72/8 - 63/8
2k m 9/8
Since the plane equation must hold, we can solve for the values of k and m. We simplify to find the parameter value for the projection point B.
k -1, m 17/4
Substituting these values back into the parametric equations of the line gives the coordinates of the projection point:
x1 -1, y 0, z 5
So, the projection point on the plane is B(0, -2, 2).
4. Equation of the Projected Line AB'
The projected line AB' is defined by the direction vector AB', which is the difference between the coordinates of points A and B'.
AB' (0 - 7/8, -2 - (-7/4), 2 - 19/8) (-7/8, -1/4, -3/8)
Normalize the direction vector:
(-2, 4, 10) — (-1, 2, 5)
Therefore, the parametric equations of the projected line AB' are:
x - 2/-1 y 6/2 z 8/5
This is the equation of the projected line on the plane.
Conclusion
The process of finding the projection of a line on a plane and the intersection of lines with planes involves a series of complex calculations and conceptual understanding. By following the step-by-step approach outlined in this article, one can effectively solve similar problems in analytic geometry.