Finding Points on Line / Intercepts of Line

Lines in 2D

The Standard Form of equation of Line in 2D is given as

\(Ax + By + C=0\)

The following are the steps to find the Points on the Line.

1. If both co-efficients \(A\) and \(B\) are not zero, then any arbitrary value can be put for any coordinate to find the other coordinate.
2. If any one of the two co-efficient is missing, then the line is either parallel to that coordinate (when \(C\neq0\)) or the line contains that coordinate (when \(C=0\)) and the coordinate can have any value for the Line. The other coordinate will be a constant and the Line is perpendicular to that coordinate axis.

1. If \(C=0\) then the Line passes through the origin and does not have any other intercepts.
2. If \(C\neq0\) then intercept of any axis can be found by setting the other coordinate value to 0 and then evaluating the non-zero coordinate axis.

Lines in 3D

The Point Direction Ratio Form of scalar equation of a Line in 3D passing through point \((x_1,y_1,z_1)\) having a direction ratio \((x_d,y_d,z_d)\) is given as following:

\(\frac{x-x_1}{x_d}=\frac{y-y_1}{y_d}=\frac{z-z_1}{z_d}\)

The following are the steps to find the Points on the Line.

1. Resolve the above equation into sets of 2 explicit equations of planes each as given in the following:
   
   \(y = Ax + B , \hspace{.5cm} z = Cx + D\)     (When \(y\) and \(z\) are given as a function of \(x\))
   
   \(z = Ay + B , \hspace{.5cm} x = Cy + D\)     (When \(z\) and \(x\) are given as a function of \(y\))
   
   \(x = Az + B , \hspace{.5cm} y = Cz + D\)     (When \(x\) and \(y\) are given as a function of \(z\))
2. Using any of the explicit equation set, the value of the dependent coordinates can be found out by putting any arbirary value of the independent coordinate in the equations. For example, when \(y\) and \(z\) are given as a function of \(x\), any arbitrary value of the \(x\) can be put to find the value of \(y\) and \(z\).

1. To find the Intercept of the line on \(xy\) plane put the value of the \(z\) coordinate to 0 in the Explicit Equation Set where \(x\) and \(y\) are given as a function of \(z\)
2. To find the Intercept of the line on \(yz\) plane put the value of the \(x\) coordinate to 0 in the Explicit Equation Set where \(y\) and \(z\) are given as a function of \(x\).
3. To find the Intercept of the line on \(xz\) plane put the value of the \(y\) coordinate to 0 in the Explicit Equation Set where \(x\) and \(z\) are given as a function of \(y\).