Are the lines L1 and L2 the same line? L1: x=1+2 t, y=1-3 t, z=1+t L2: x=1-4 t, y=6 t

step 1 So are these two lines the same? Let's rewrite them to separate their base point and their line

Are the lines L1 and L2 the same line? L1: x=1+2 t,...

Key Concepts

Parametric Equations of a Line

A line in space is often described using parametric equations where each coordinate is expressed as a function of a parameter. This representation includes a fixed point (the initial position) and a direction vector that indicates the line’s direction. The parametric form is a powerful tool in vector geometry because it allows for straightforward computations of points along the line, intersections, and relationships with other lines or geometric objects.

Direction Vectors

The direction vector of a line in its parametric representation is extracted from the coefficients of the parameter in the coordinate equations. This vector defines the orientation of the line in space. In problems involving the analysis of lines, comparing direction vectors is fundamental to determining whether lines are parallel, intersect, or, in some cases, identical.

Coincident Lines

Two lines are considered to be coincident (i.e., the same line) if they share all points, which occurs when one line’s points lie entirely on the other. To check for coincidence, one must compare both the direction vectors to verify they are parallel and then assess whether a point on one line can be expressed as a point on the other line using the respective parameters. This involves solving for the parameters to ensure consistency across all coordinate equations.

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