Find parametric equations for the line through P1 and P2 and...

Find parametric equations for the line through $P_{1}$ and $P_{2}$ and also for the line segment joining those points.
$$
\begin{array}{l}{\text { (a) } P_{1}(0,1), \quad P_{2}(-3,-4)} \\ {\text { (b) } P_{1}(-1,3,5), \quad P_{2}(-1,3,2)}\end{array}
$$

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Key Concepts

Parametric Equations

Parametric equations express the coordinates of the points on a line as functions of one or more parameters. This involves representing each coordinate as an initial point plus a scalar multiple of a direction vector, allowing the description of the line in a concise and flexible manner in the given dimension.

Direction Vector

The direction vector of a line is obtained by subtracting the coordinates of one point from another point on the line. This vector indicates the orientation and magnitude of the line's slope in the space, forming a fundamental component in constructing the parametric equations.

Line Segment vs Infinite Line

The line defined by parametric equations can extend infinitely in both directions when the parameter is any real number. However, a line segment is the portion of the line that lies between two specific points; therefore, its parametric representation must restrict the parameter to a closed interval that corresponds to those endpoints.

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