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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

\[\frac{x-{{x}_{1}}}{l}=\frac{y-{{y}_{1}}}{m}=\frac{z-{{z}_{1}}}{n}\]

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

\[\frac{x-{{x}_{1}}}{{{x}_{2}}-{{x}_{1}}}=\frac{y-{{y}_{1}}}{{{y}_{2}}-{{y}_{1}}}=\frac{z-{{z}_{1}}}{{{z}_{2}}-{{z}_{1}}}\]

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

\[\frac{x-{{x}_{1}}}{l}=\frac{y-{{y}_{1}}}{m}=\frac{z-{{z}_{1}}}{n}=r\]

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.