Are the lines L1 and L2 the same line? L1: x=2+3 t, y=1+4 t, z=6-t L2: x=5+6 t, y=5+8

step 1 We want to see if these two lines are the same. To that end, I'm going to write them out in a, b

Are the lines L1 and L2 the same line? L1: x=2+3 t,...

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Key Concepts

Coincident Lines

For two lines to be coincident, they must not only be parallel (having proportional direction vectors) but also share at least one common point. This involves verifying that a point from one line satisfies the parametric equations of the other, ensuring that every point on one line corresponds to a point on the other.

Solving Systems of Equations

To confirm whether two lines are the same, one typically sets up a system by equating the corresponding parametric components. Solving this system helps determine if there exists a consistent parameter mapping that relates the two sets of parametric equations, thereby establishing whether the lines coincide.

Parametric Representation of Lines

A parametric representation describes a line by specifying equations for each coordinate in terms of a parameter. This form clearly shows a point on the line and the direction in which the line extends, making it a versatile tool to analyze the properties and relationships between lines in space.

Direction Vectors

The direction vector, extracted from the parametric equations, defines the orientation of a line. Comparing the direction vectors of two lines is a fundamental step in determining whether they are parallel or have the potential to be identical, as they must be proportional for the lines to have the same orientation.

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