Intersecting lines Consider the following pairs of lines. Determine whether the lines are parallel or intersecting. If the lines intersect, then determine the point of intersection.


c. x = 1 + 3s, y = 4 + 2s and x = 4 - 3t, y = 6 + 4t

Verified step by step guidance

1. Identify the parametric equations of the two lines: Line 1 is given by \(x = 1 + 3s\), \(y = 4 + 2s\) and Line 2 is given by \(x = 4 - 3t\), \(y = 6 + 4t\), where \(s\) and \(t\) are parameters.
2. To check if the lines are parallel, compare their direction vectors. The direction vector of Line 1 is \(\langle 3, 2 \rangle\) and for Line 2 it is \(\langle -3, 4 \rangle\). Determine if one vector is a scalar multiple of the other.
3. If the direction vectors are not scalar multiples, the lines are not parallel and may intersect. To find the intersection point, set the \(x\) and \(y\) coordinates equal to each other: \$1 + 3s = 4 - 3t\( and \)4 + 2s = 6 + 4t$.
4. Solve the system of equations for \(s\) and \(t\). This involves rearranging the equations and using substitution or elimination methods to find values of \(s\) and \(t\) that satisfy both equations simultaneously.
5. Once you find \(s\) and \(t\), substitute either value back into the parametric equations of one of the lines to find the coordinates of the intersection point \(\left(x, y\right)\).

Key Concepts

Parametric Equations of Lines

Parametric equations express the coordinates of points on a line as functions of a parameter, typically denoted by s or t. Each parameter value corresponds to a unique point on the line, allowing a clear representation of lines in the plane or space.

Determining Intersection of Lines

To find if two lines intersect, set their parametric equations equal and solve for the parameters. If a consistent solution exists, the lines intersect at the corresponding point; otherwise, they are parallel or skew.

Parallel Lines and Direction Vectors

Two lines are parallel if their direction vectors are scalar multiples of each other. Comparing the coefficients of the parameters in the parametric equations helps identify parallelism without solving for intersection.