A total of 50 types of lines can be laid out in a Cartisean Space in 3D divided into 6 Categories. Following are they:

- Lines that a perpendicular to a coordinate planes. There are 3 types of such lines.
S.No Equation **Explanation**1. \( x=D \)

\( y=D \)These lines are perpendicular to *XY plane*(and consequently parallel to*Z axis*).2. \( y=D \)

\( z=D \)These lines are perpendicular to *YZ plane*(and consequently parallel to*X axis*).3. \( x=D \)

\( z=D \)These lines are perpendicular to *XZ plane*(and consequently parallel to*Y axis*). - Lines that a pass through origin but are not parallel to any plane. There are 4 types of such lines.
**S.No****Equation****Explanation**1. \(\frac{x}{A} = \frac{y}{B} = \frac{z}{C}\) *OR*

\(\frac{-x}{A} = \frac{-y}{B} = \frac{-z}{C}\)These lines are aligned in 1st and 7th octant 2. \(\frac{x}{A} = \frac{y}{B} = \frac{-z}{C}\) *OR*

\(\frac{-x}{A} = \frac{-y}{B} = \frac{z}{C}\)These lines are aligned in 2nd and 8th octant 3. \(\frac{x}{A} = \frac{-y}{B} = \frac{z}{C}\) *OR*

\(\frac{-x}{A} = \frac{y}{B} = \frac{-z}{C}\)These lines are aligned in 3rd and 5th octant 4. \(\frac{-x}{A} = \frac{y}{B} = \frac{z}{C}\) *OR*

\(\frac{x}{A} = \frac{-y}{B} = \frac{-z}{C}\)These lines are aligned aligned in 4th and 6th octant - Lines that are parallel to or lie on a coordinate plane and pass through the axis perpendicular to that plane. There are 6 types of such lines
**S.No****Equation****Explanation**1. \(\frac{x}{A} = \frac{y}{B}\) *OR*

\(\frac{-x}{A} = \frac{-y}{B}\)*AND*

\(z=D \)These lines lie on or are parallel to *XY plane*and are perpendicular to and pass through*Z axis*2. \(\frac{-x}{A} = \frac{y}{B}\) *OR*

\(\frac{x}{A} = \frac{-y}{B}\)*AND*

\(z=D \)These lines lie on or are parallel to *XY plane*and are perpendicular to and pass through*Z axis*3. \(\frac{y}{B} = \frac{z}{C}\) *OR*

\(\frac{-y}{B} = \frac{-z}{C}\)*AND*

\(x=D \)These lines lie on or are parallel to *YZ plane*and are perpendicular to and pass through*X axis*4. \(\frac{-y}{B} = \frac{z}{C}\) *OR*

\(\frac{y}{B} = \frac{-z}{C}\)*AND*

\(x=D \)These lines lie on or are parallel to *YZ plane*and are perpendicular to and pass through*X axis*5. \(\frac{x}{A} = \frac{z}{C}\) *OR*

\(\frac{-x}{A} = \frac{-z}{C}\)*AND*

\(y=D \)These lines lie on or are parallel to *XZ plane*and are perpendicular to and pass through*Y axis*6. \(\frac{-x}{A} = \frac{z}{C}\) *OR*

\(\frac{x}{A} = \frac{-z}{C}\)*AND*

\(y=D \)These lines lie on or are parallel to *XZ plane*and are perpendicular to and pass through*Y axis* - Lines that are parallel to or lie on a coordinate plane and are perpendicular to axis perpendicular to that plane but do not pass through it. There are 12 types of such lines 4 for each coordinate plane.
**S.No****Equation****Intercepts****Explanation**1. \(\frac{x-x1}{A} = \frac{y-y1}{B}\) *AND*

\(z=D \)XZ Plane intercept point =(\(\frac{-Ay1+Bx1}{B},0,D\))

YZ Plane intercept point =(\(0,\frac{-Bx1+Ay1}{A},D\))Lines perpendicular to *Z axis*either lying on or parallel to XY plane, having intercepts on XZ and YZ plane. 4 subtypes of such lines types are possible2. \(\frac{y-y1}{B} = \frac{z-z1}{C}\) *AND*

\(x=D \)XY Plane intercept point =(\(D,\frac{-Bz1+Cy1}{C},0\))

XZ Plane intercept point =(\(D,0,\frac{-Cy1+Bz1}{B}\))Lines perpendicular to *X axis*either lying on or parallel to YZ plane, having intercepts on XY and XZ plane. 4 subtypes of such lines types are possible3. \(\frac{x-x1}{A} = \frac{z-z1}{C}\) *AND*

\(y=D \)XY Plane intercept point =(\(\frac{-Az1+Cx1}{C},D,0\))

YZ Plane intercept point =(\(0,D,\frac{-Cx1+Az1}{A}\))Lines perpendicular to *Y axis*either lying on or parallel to XZ plane, having intercepts on XY and YZ plane. 4 subtypes of such lines types are possible - Lines that pass through a coordinate axis and form intercept on a coordinate plane not involving that axis. There are 24 types of such lines 8 for each coordinate axis/plane.
**S.No****Equation****Intercept****Explanation**1. \(\frac{x}{A}=\frac{y-y1}{B}=\frac{z}{C}\) XZ Plane intercept point =(\(\frac{-Ay1}{B},0,\frac{-Cy1}{B}\)) These line pass through *Y axis*and form intercept on*XZ plane*. 8 subtypes of such lines types are possible, 4 for negative values of y1 and 4 for positive values of y1.2. \(\frac{x-x1}{A}=\frac{y}{B}=\frac{z}{C}\) YZ Plane intercept point =(\(0,\frac{-Bx1}{A},\frac{-Cx1}{A}\)) These line pass through *X axis*and form intercept on*YZ plane*. 8 subtypes of such lines types are possible, 4 for negative values of x1 and 4 for positive values of x13. \(\frac{x}{A}=\frac{y}{B}=\frac{z-z1}{C}\) XY Plane intercept point =(\(\frac{-Az1}{C},\frac{-Bz1}{C},0\)) These line pass through *Z axis*and form intercept on*XY plane*. 8 subtypes of such lines types are possible, 4 for negative values of z1 and 4 for positive values of z1 - Lines that form non zero intercepts on all three coordinate planes
**S.No****Equation****Intercept****Explanation**1. \(\frac{x-x1}{A}=\frac{y-y1}{B}=\frac{z-z1}{C}\) XY Plane intercept point =(\(\frac{-Az1+Cx1}{C},\frac{-Bz1+Cy1}{C},0\))

YZ Plane intercept point =(\(0,\frac{-Bx1+Ay1}{A},\frac{-Cx1+Az1}{A}\))

XZ Plane intercept point =(\(\frac{-Ay1+Bx1}{B},0,\frac{-Cy1+Bz1}{B}\))These lines make intercept with all three coordinate planes