Equations of Three Dimensional Lines

Go back to  'Three Dimensional Geometry'

In this section, we’ll discuss how to write the equation for a straight line in coordinate form. There are essentially two different ways of doing so:

UNSYMMETRICAL FORM OF  THE EQUATION OF A LINE : A line can be defined as the intersection of two planes. Thus, the equations of two planes considered together represents a straight line. For example, the set of equations

\[\begin{align}  & {{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}z+{{d}_{1}}=0 \\\\  & {{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}z+{{d}_{2}}=0 \\ \end{align}\]

represents the straight line formed by the intersection of these two planes. Recall that the planes will intersect only if they are non-parallel, i.e., only if

\[{{a}_{1}}:{{b}_{1}}:{{c}_{1}}\ne {{a}_{2}}:{{b}_{2}}:{{c}_{2}}\]

SYMMETRICAL FORM OF  THE EQUATION OF A LINE : Consider a line with direction cosines l, m, n and passing through the point  \(A({{x}_{1}},{{y}_{1}},{{z}_{1}}).\) For any point  \(P\left( x,y,z \right)\) on this line, the set of numbers  \(\left\{ \left( x-{{x}_{1}} \right),\left( y-{{y}_{1}} \right),\left( z-{{z}_{1}} \right) \right\}\) must be proportional to the direction cosines, as has already been discussed. Thus, the equation of this line can be written as


Extending this, we can write the equation of the line passing through  \(A\left( {{x}_{1}},{{y}_{1}},{{z}_{1}} \right)\) and  \(B\left( {{x}_{2}},{{y}_{2}},{{z}_{2}} \right)\) as


Note that for any point  \(P\left( x,y,z \right)\)  at a distance r from  \(A\left( {{x}_{1}},{{y}_{1}},{{z}_{1}} \right)\)  along the line with direction cosines l, m, n, we have


Thus, the coordinates of P can be written as

\[\boxed{x = {x_1} + lr,\,\,\,\,\,y = {y_1} + mr,\,\,\,\,\,z = {z_1} + nr}\]

This is a useful fact and we’ll be using it frequently.