Vector and Parametric Equations of a Line in 3D

Find a vector equation and parametric equations for the line that passes through the point $(5, 1, 3)$ and is parallel to the vector $\mathbf{v} = (1, 4, -2)$. Then find two other points on the line.

In this problem, we explore the concept of representing a line in three-dimensional space through vector and parametric equations. Understanding these representations is crucial because they provide precise mathematical ways to describe a line's orientation and position in space. A vector equation of a line makes use of a position vector and a direction vector. The direction vector indicates the line's orientation, while the position vector indicates a specific point through which the line passes.

Parametric equations, on the other hand, parameterize the coordinates of the points on the line, often using a parameter, typically denoted as 't'. These equations allow us to express each coordinate of the points on the line as a linear function of the parameter. By manipulating this parameter, you can obtain any point along the line, giving flexibility and utility in applications such as computer graphics and physics simulations.

In this problem, not only do you find the equations of the line based on given point and vector, but you also have the opportunity to find additional points on the line, showing the relationship between the vector direction and the positioning of points along the line. This strengthens understanding of how lines in three dimensions can be navigated using these mathematical tools.

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