JEE Main & Advanced Mathematics Three Dimensional Geometry Intersection of Two Lines

Determine whether two lines intersect or not. In case they intersect, the following algorithm is used to find their point of intersection.

Algorithm:

Let the two lines be  \[\frac{x-{{x}_{1}}}{{{a}_{1}}}=\frac{y-{{y}_{1}}}{{{b}_{1}}}=\frac{z-{{z}_{1}}}{{{c}_{1}}}\]             …..(i)

and  \[\frac{x-{{x}_{2}}}{{{a}_{2}}}=\frac{y-{{y}_{2}}}{{{b}_{2}}}=\frac{z-{{z}_{2}}}{{{c}_{2}}}\]                        …..(ii)

Step I : Write the co-ordinates of general points on (i) and (ii). The co-ordinates of general points on (i) and (ii) are given by \[\frac{x-{{x}_{1}}}{{{a}_{1}}}=\frac{y-{{y}_{1}}}{{{b}_{1}}}=\frac{z-{{z}_{1}}}{{{c}_{1}}}=\lambda \] and \[\frac{x-{{x}_{2}}}{{{a}_{2}}}=\frac{y-{{y}_{2}}}{{{b}_{2}}}=\frac{z-{{z}_{2}}}{{{c}_{2}}}=\mu \] respectively.

i.e., \[({{a}_{1}}\lambda +{{x}_{1}},\,{{b}_{1}}\lambda +{{y}_{1}}+{{c}_{1}}\lambda +{{z}_{1}})\]and \[({{a}_{2}}\mu +{{x}_{2}},\,{{b}_{2}}\mu +{{y}_{2}},\,{{c}_{2}}\mu +{{z}_{2}})\].

Step II : If the lines (i) and (ii) intersect, then they have a common point. \[{{a}_{1}}\lambda +{{x}_{1}}={{a}_{2}}\mu +{{x}_{2}},\,{{b}_{1}}\lambda +{{y}_{1}}={{b}_{2}}\mu +{{y}_{2}}\] and \[{{c}_{1}}\lambda +{{z}_{1}}={{c}_{2}}\mu +{{z}_{2}}\].

Step III : Solve any two of the equations in \[\lambda \] and \[\mu \] obtained in step II. If the values of l and m satisfy the third equation, then the lines (i) and (ii) intersect, otherwise they do not intersect.

Step IV : To obtain the co-ordinates of the point of intersection, substitute the value of \[\lambda \] (or \[\mu \]) in the co-ordinates of general point (s) obtained in step I.