JEE Main & Advanced Mathematics Three Dimensional Geometry Straight Line in Space

Every equation of the first degree represents a plane. Two equations of the first degree are satisfied by the co-ordinates of every point on the line of intersection of the planes represented by them.

Therefore, the two equations of that line \[ax+by+cz+d=0\] and \[{a}'x+{b}'y+{c}'z+{d}'=0\] together represent a straight line.

(1) Equation of a line passing through a given point

Cartesian equation of a straight line passing through a fixed point \[({{x}_{1}},\,{{y}_{1}},\,{{z}_{1}})\] and having direction ratios \[a,b,c\] is \[\frac{x-{{x}_{1}}}{a}=\frac{y-{{y}_{1}}}{b}=\frac{z-{{z}_{1}}}{c}\].

(2) Equation of line passing through two given points

If \[A({{x}_{1}},\,{{y}_{1}},\,{{z}_{1}}),\,B({{x}_{2}},\,{{y}_{2}},\,{{z}_{2}})\] be two given points, the equations to the line AB are \[\frac{x-{{x}_{1}}}{{{x}_{2}}-{{x}_{1}}}=\frac{y-{{y}_{1}}}{{{y}_{2}}-{{y}_{1}}}=\frac{z-{{z}_{1}}}{{{z}_{2}}-{{z}_{1}}}\].