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In mathematics, an intersection is a point or a set of points where two or more geometric entities meet. In the context of coordinate planes, intersections are where a line or a curve crosses one of the planes.
When dealing with a line defined by parametric equations, the goal is to determine where this line intersects each of the coordinate planes. Let's first understand the basic planes:

• XY-Plane: Defined by setting the z-coordinate to zero.
• XZ-Plane: Defined by setting the y-coordinate to zero.
• YZ-Plane: Defined by setting the x-coordinate to zero.

To find these intersections, we substitute for zero in the non-relevant coordinate and solve for the parameter. This parameter is then used to find the remaining coordinates on the intersection point.
This method helps visualize how lines interact with these planes, offering deeper insights into the geometry of three-dimensional space.

Line equations on a coordinate plane can come in various forms. For three-dimensional space, lines are often expressed parametrically. In parametric equations, each coordinate (x, y, and z) of a point on the line is expressed as a function of a single parameter, usually denoted as \( t \).
For instance, in the line equation from our problem:

• \( x = 1 + 2t \)
• \( y = -1 - t \)
• \( z = 3t \)

These equations indicate how each coordinate changes as \( t \) changes.

• The term in front of the parameter \( t \) represents the direction vector of the line.
• The constant term shows a particular point through which the line passes when \( t = 0 \).

By understanding these terms, one can easily derive how the line behaves and intersects different planes.

The Cartesian coordinate system is a system that uses coordinates to uniquely determine the position of a point in space. It comprises axes that intersect at a point called the origin, usually denoted as \( (0, 0, 0) \) in three-dimensional space.
In three dimensions, these axes are:

• The x-axis: Extends horizontally.
• The y-axis: Extends vertically.
• The z-axis: Extends into and out of the plane of x and y.

Points are defined as \( (x, y, z) \), where each value corresponds to the position along the respective axis.
Understanding how to plot and interpret points in this system is crucial for solving geometry problems, such as finding intersections with coordinate planes. It establishes a fundamental understanding of three-dimensional spatial relationships, simplifying complex spatial reasoning.