Determine whether the lines l, and l2 are parallel, skew or...

Determine whether the lines $l$, and $l_{2}$ are parallel, skew or intersecting. If they intersect, find the coordinates of the point of intersection.
$$\frac{x-2}{5}=y-1=\frac{z-2}{3} \text { and } \frac{x+4}{3}=\frac{7-y}{3}=\frac{10-z}{4}$$

Key Concepts

Parametric Representation of Lines

This concept involves expressing a line in space using parameters, typically by setting up equations that indicate how each coordinate (x, y, z) varies linearly with a parameter. It is a fundamental method to describe lines, making it easier to compare lines and find points of intersection.

Direction Vectors

A direction vector represents the orientation of a line in space and is derived from the coefficients of the parameter in the parametric equations. It is crucial for determining whether two lines are parallel, as lines are parallel if their direction vectors are scalar multiples of each other.

Intersection of Lines

Determining the intersection of lines in three-dimensional space involves equating the corresponding parametric equations and solving the resulting system of equations. If a common solution exists, it represents the point at which the two lines meet.

Skew Lines

Skew lines are lines that do not intersect and are not parallel. Recognizing skew lines requires understanding that, even if the lines do not have parallel direction vectors, there may not exist any point that satisfies both sets of parametric equations simultaneously.

System of Equations

When finding the intersection point of two lines, one sets up a system of linear equations derived from the parametric representations of the lines. Solving this system can show whether a common solution (i.e., point of intersection) exists or if the lines are skew.

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