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Linear algebra is a branch of mathematics concerned with the study of vectors, vector spaces, linear transformations, and systems of linear equations. It provides a framework for solving a wide range of mathematical problems and is fundamental to many areas of math and engineering.

The key objects of study in linear algebra are vectors, which are abstract entities that can be added together and multiplied by scalars to produce new vectors. These operations must satisfy certain axioms, leading to what we call a vector space. Linear transformations are functions between vector spaces that preserve vector addition and scalar multiplication. The coordinate systems that we work with, represented by matrices, are a concrete representation of these concepts, and allow us to perform calculations and solve problems.

A system of equations is a collection of two or more equations with a set of variables. Solving the system means finding values for these variables that satisfy all equations simultaneously. This task pervades linear algebra, where systems often represent intersections of geometric objects, like lines and planes.

Systems can be solved using a variety of methods, including substitution, elimination, and matrix operations. In the context of finding the intersection between two planes, we set their equations equal to each other and solve for the variables, which provides a set of coordinates. These coordinates represent points common to both planes, and thus sit on the line of intersection if it exists.

The cross product is a special operation in three-dimensional space involving two vectors. Its result is a new vector that is perpendicular to both of the original vectors, making this operation particularly useful in problems involving orthogonality and rotation.

In the context of finding a line that is the intersection of two planes, the cross product helps identify the direction vector for that line, as this vector must be normal to both planes. The formula involves both the coefficients and the constants of the plane equations, and the result is essential for formulating the parametric equations representing the line of intersection.

A direction vector is a vector that indicates the direction of a line or vector field. In the context of a line in three-dimensional space, it tells us which way the line proceeds from any point on it. In linear algebra, the concept of direction vectors is used extensively to define lines and planes.

When given the parametric equations of a line, the coefficients of the parameter adjacent to the initial point's coordinates make up the components of the direction vector. This vector is special because it not only indicates direction but also forms the backbone of operations like finding the angle between two lines or determining if lines are parallel.

The intersection of planes in three-dimensional space can result in different geometric figures: a line, a point, or no intersection at all. When two non-parallel planes intersect, they do so along a line. Finding this line involves solving for where the planes meet — essentially finding common solutions to their equations that satisfy both plane equations simultaneously.

This concept is critical within linear algebra as it allows for the visualization and understanding of space and geometry through equations. By applying the steps outlined in the step-by-step solution, one can find both a point on the line of intersection and its direction vector, needed for the parametric equations that describe the entire line.