Determine whether the lines L1 and L2 are parallel, skew, or...

Determine whether the lines $L_{1}$ and $L_{2}$ are parallel, skew, or intersecting. If they intersect, find the point of intersection.
$L_{1}: x=5-12 t, \quad y=3+9 t, \quad z=1-3 t$
$L_{2}: x=3+8 s, \quad y=-6 s, \quad z=7+2 s$

Key Concepts

Skew Lines

Skew lines in three-dimensional space are lines that do not intersect and are not parallel. They do not lie in the same plane, and determining skewness usually involves showing that the lines are not parallel and that no solution exists for their intersection.

Intersection of Lines

To determine if two lines intersect, one sets their parametric forms equal to each other and solves the resulting system of equations for the parameters. If a consistent solution exists, then the lines share a common point, and that point is the intersection of the lines.

Parametric Representation of Lines

The parametric representation expresses a line in space as a set of equations where each coordinate (x, y, z) is given as a function of a parameter. This form clearly defines a specific point that lies on the line along with a direction vector that indicates the line's orientation in three-dimensional space.

Direction Vectors

The direction vector of a line is derived from its parametric equations as the coefficients of the parameter. This vector provides information on the direction in which the line extends and is central for determining relationships such as parallelism or intersection between lines.

Parallelism

Two lines are parallel if their direction vectors are scalar multiples of one another. Checking for parallelism involves comparing the direction vectors; if one vector can be obtained by multiplying the other by a constant, then the lines are either parallel or coincident.

Transcript

Please give Ace some feedback