91Ó°ÊÓ

When two planes intersect in three-dimensional space, they typically form a line. Visualize two sheets of paper crossing each other; their common edge represents the line of intersection. To find this line, we solve a system of plane equations simultaneously. The line exists where both plane equations hold true simultaneously.
The line can be represented in a parametric form, which involves determining a specific point on this line and a direction vector that shows its orientation. The key steps for finding this line include:

• Identifying points satisfying both plane equations, which help us find a specific location on the line.
• Determining the direction vector, which provides the line's path or orientation.

If you've ever tried to find the meeting point of two roads, you were essentially finding a line of intersection!

Plane equations describe flat surfaces in 3D space and have a general form of \( ax + by + cz = d \). Each plane equation consists of coefficients that multiply the variables \( x, y, \text{and} z \), and a constant \( d \). These coefficients determine the plane's orientation and position in space.
In exercises involving intersection, you'll usually see two plane equations, similar to the originals given: \( x + y + z = 1 \) and \( x + y = 2 \). Here's how to understand them:

• The first equation implies a three-dimensional surface, where any point \( (x, y, z) \) satifying it is on the plane.
• The second equation can be visualized as a tilted plane where every combination of \( (x, y, z) \) that makes it true lies on this plane.

Finding their intersection line involves checking which combinations of \( x, y, \) and \( z \) satisfy both equations simultaneously.

A direction vector is crucial in defining the line of intersection between two planes. It indicates the line's orientation and direction in three-dimensional space. Think of it as an arrow showing the line's path.
To determine the direction vector in such problems, we typically parameterize one of the variables, like \( z = t \), and solve the plane equations with this parameter. Once you find how the other variables relate to \( t \), you'll notice a pattern or a direction vector.
In the exercise, the direction vector \( (0, -1, 1) \) was found by observing how \( x \), \( y \), and \( z \) change as \( t \) varies. This vector suggests that moving one unit positively in the \( z \)-direction accompanies a one-unit negative change in the \( y \)-direction, while \( x \) remains constant.
This part of the exercise highlights the beauty of geometry: a set of numbers beautifully describes how lines behave and extend in space.