91Ó°ÊÓ

When two planes intersect in three-dimensional space, their intersection forms a line. To identify this line from the planes given by the equations \( x + y + t = 0 \) and \( x - t = 0 \), we solve these equations together. This means we look for the set of points that satisfy both. The second equation, \( x - t = 0 \), simplifies the determination of \( x \), allowing us to state that \( x = t \).
Using this expression for \( x \) in the first equation provides further simplifications. By substituting \( x = t \), the first equation becomes \( t + y + t = 0 \), which simplifies to \( 2t + y = 0 \). Solving these two equations simultaneously enables us to describe the line where these planes intersect. This intersection line can be visualized as a collection of all possible solutions that meet both plane equations.
Hence, finding the intersection of two planes essentially involves solving their simultaneous equations for a set of linear solutions.

The direction vector helps us express both the orientation and extent of a straight line in space. For the line of intersection described by the simultaneous equations, the direction is rendered through a vector. In our previous example, after determining that \( x = t \) and \( y = -2t \), the line can be represented with the parametric form \( (x, y, t) = (t, -2t, t) \).
This form allows us to identify the direction vector, which is \( \begin{pmatrix} 1 \ -2 \ 1 \end{pmatrix} \). Such a vector specifies the changes in \( x \), \( y \), and \( t \) as we move along the line, effectively showing where and how fast each coordinate changes.
Understanding the direction vector is critical in creating a projection matrix for projecting points onto a line, a common application in computer graphics and data analysis. Essentially, it's the backbone of defining a line's spatial trajectory.

The parametric representation of a line is a simple and effective way to express a line in three-dimensional space. Derived from the equations of the planes \( x + y + t = 0 \) and \( x - t = 0 \), the parametric form we obtained was \( (x, y, t) = (t, -2t, t) \).
This form uses a parameter, often denoted by \( t \), that can take any real value to determine specific points on the line. Each component of the vector equation shows the relationship between a point on the line and the parameter \( t \).
The beauty of this representation lies in its simplicity, enabling us to visualize and calculate exact points for different values of \( t \), making it easier to apply transformations like projection or determine where the line intersects with other geometrical objects.

Simultaneous equations play a crucial role when determining the intersection line between two planes. By solving them, we essentially find all the points that lie on both planes. For the given planes \( x + y + t = 0 \) and \( x - t = 0 \), we used simultaneous equations to simplify and eventually eliminate variables to express the intersection in terms of a single parameter \( t \).
The technique involves expressing one variable in terms of another, which we did by letting \( x = t \). Plugging this into the first equation, we found \( y = -2t \).
This process allows mathematicians and engineers to define constraints and relationships between multiple equations quickly. Simultaneous equations are a powerful mathematical tool, foundational in algebra, calculus, and applied mathematics, helping us to handle a plethora of practical problems from high school physics to cutting-edge engineering.