Final answer:
Solutions to a system of equations in parametric vector form are expressed through vectors representing particular solutions and directions. The geometric comparison involves studying the consistency and relation between solutions sets. Projectile problems are solved by breaking down motion into horizontal and vertical components.
Step-by-step explanation:
To describe the solutions of the first system of equations in parametric vector form, you need to express the solution set as a vector plus a parameter times another vector. This typically involves adding and subtracting equations to eliminate variables and express one variable in terms of another. Given a system of linear equations, the solution in parametric vector form will consist of a fixed vector (representing a particular solution) plus a parameter times a direction vector (representing the direction and magnitude of the line that constitutes the solution set).
The geometric comparison with the solution set of the second system of equations involves determining if the systems are consistent and if they result in intersecting lines (a single solution), parallel lines (no solution), or coincident lines (infinite solutions). Two-dimensional vector problems are analyzed by breaking them down into their horizontal (x-axis) and vertical (y-axis) components, which simplifies calculations and visualizations.
To solve projectile problems, you separate the motion into horizontal and vertical components, solve for each direction, and then combine the components using vector addition equations. This approach applies to any two-dimensional motion problem where the use of a coordinate system parallel to one or more of the vectors simplifies the mathematics.