Non Coplanar Points

The points, which do not lie in the same geometrical plane, are known as non-coplanar points. Any 3 points can be enclosed by one geometrical plane, but four or more points cannot always be enclosed by one geometrical plane. If the points belong to the same plane then they are known as coplanar points. In this section, we shall discuss the non coplanar points.

 

Example for non coplanar points:

 

nonCoplanar

  • From tha above shown figure the points A , B , N , P are non coplanar as they do not lie on the same plane.

  • We can determine four planes with the help of four non co-planar points.

  • Plane is nothing but the two dimensional geometrical object .

 

Solved example for non coplanar points:

 

Example Problem 1:

Check whether the following lines are co-planar or not.

 2x+6y+9 = 0 and 3x+4y+11 = 0

Solution:

  • The given equations are  2x+6y+9 = 0 and 3x+4y+11 = 0

  • The slope intercept form can be given as y = mx+b

  • Where m indicates slope.

  • Comparing the above equation with the given equation, we get:

  • 6y = -2x-9  

  •  Dividing by 6 on both sides we get:

  • y = `-2/6x-9/6`

 

  • We get m = `-2/6 = -1/3` ------- (1)

  • The slope intercept form can be given as y = nx+b

  • Where n indicates slope.

  • Comparing the above equation with the given equation, we get:

  • 4y = -3x-11  

  •  Dividing by 4 on both sides we get:

  • y = `-3/4x-11/4`

 

  • We get n = `-3/4` --------- (2)

  • Equation (1) `!=` (2), that is m `!=` n

  • That is the slopes of the two equations are not equal (that is the lines are not parallel) and therefore the points lie on the two lines are non co-planar points.

     

Example Problem 2:

Check whether the following lines are co-planar or not.

 x+5y+9 = 0 and 2x+10y+11 = 0

Solution:

  • The given equations are  x+5y+9 = 0 and 2x+10y+11 = 0

  • The slope intercept form can be given as y = mx+b

  • Where m indicates slope.

  • Comparing the above equation with the given equation, we get:

  • 5y = -x-9  

  •  Dividing by 5 on both sides we get:

  • y = `-1/5x-9/5`

 

  • We get m = `-1/5 ` ------- (1)

  • The slope intercept form can be given as y = nx+b

  • Where n indicates slope.

  • Comparing the above equation with the given equation, we get:

  • 10y = -2x-11  

  •  Dividing by 10 on both sides we get:

  • y = `-2/10x-11/10`

 

  • We get n = `-2/10 = -1/5` --------- (2)

  • Equation (1) `=` (2), that is m `=` n

That is the slopes of the two equations are equal (that is the lines are  parallel) and therefore the points lie on the two lines are co-planar points.