The Line \(L_1\) is represented by Position Vector \(\vec{L_1}\) and it Passes through Point having Position Vector \(\vec{P_1}\) having a Direction given by Vector \(\vec{D_1}\).
The Line \(L_2\) is represented by Position Vector \(\vec{L_2}\) and it Passes through Point having Position Vector \(\vec{P_2}\) having a Direction given by Vector \(\vec{D_2}\).
\(t\) and \(s\) are Real Numbers that Scale the Direction Vectors \(\vec{D_1}\) and \(\vec{D_2}\) respectively.
The following gives the Steps to Find the Relation between Lines \(L_1\) and \(L_2\)
For 3D Space, the Position Vector Equations of Lines \(L_1\) and \(L_2\) can be written as
\(x_{P_1},y_{P_1},z_{P_1} = \) Components Of the Position Vector \(\vec{P_1}\)
\(x_{D_1},y_{D_1},z_{D_1} = \) Components Of the Direction Vector \(\vec{D_1}\)
\(x_{P_2},y_{P_2},z_{P_2} = \) Components Of the Position Vector \(\vec{P_2}\)
\(x_{D_2},y_{D_2},z_{D_2} = \) Components Of the Direction Vector \(\vec{D_2}\)
Now, at the Point of Intersection of 2 Lines, the Position Vector of Both \(L_1\) and \(L_2\) will be Same. Therefore by equating equations (3) and (4) we get
Similarly, In \(N\) Dimensions (where \(N \geq 2\)) , \(N\) equations of 2 Variables \(t\) and \(s\) can be obtained by equating the Position Vector Equations of the Lines \(L_1\) and \(L_2\).
If the System of Linear Equations thus obtained has No Solutions, then the Lines are Parallel if Angle between the Lines is 0.
Otherwise the Lines Lie on Different Planes.
If the System of Linear Equations thus obtained has Infinitely Many Solutions for Variables \(t\) and \(s\), then the Lines are Coincident.
If the System of Linear Equations thus obtained has a Unique Solution for Variables \(t\) and \(s\), then the Lines Intersect at a Single Point.
The Position Vector of Point of Intersection can be found by Putting the Value of \(t\) in equation (1) or by Putting the Value of \(s\) in equation (2).
In 2D, the following gives Explicit Coordinate Equations of 2 Lines \(L_1\) and \(L_2\)
\(L_1:\hspace{2mm}A_1x + B_1y = C_1\) ...(10)
\(L_2:\hspace{2mm}A_2x + B_2y = C_2\) ...(11)
The Relation between Lines \(L_1\) and \(L_2\) can be found by Finding the Solution to the System of Linear Equations given by the equations (10) and (11)
If the System of Linear Equations has No Solutions, then the Lines are Parallel.
If the System of Linear Equations has Infinitely Many Solutions for Variables \(x\) and \(y\), then the Lines are Coincident.
If the System of Linear Equations has a Unique Solution for Variables \(x\) and \(y\), then the Lines Intersect at a Single Point whose Coordinate Points are given by
Values of \(x\) and \(y\).