Intersection of Planes

Intersection of Two Planes

 

Determine whether each of the following systems of equations is consistent or inconsistent.  If the system is consistent, then state the equation of the line of intersection or show that the planes are coincident.

 

Example 1

2x + 3y - 4z = 7

6x + 9y - 12z = 22                Answer: inconsistent system; two parallel non-coincident planes                            Video Solution

 

Example 2

3x + y + 8z = 23

6x + 2y + 16z = 46                Answer: consistent system; two coincident planes                            Video Solution

 

Example 3

x - 4y + 3z = 4

2x +y - z = 6                Answer: consistent system; intersect along the line  r = (22/7 , 0 , 2/7) + t (1, 7, 9), t E R                         Video Solution

 

 

Intersection of Three Planes

 

Determine whether each of the following systems of equations is consistent or inconsistent.  If the system is consistent, then state the single point of intersection or the equation of the line of intersection or show that the planes are coincident.

 

Example 1

3x - y + 4z + 8 = 0

2x + y + z + 7 = 0

2x + 5y - 2z + 9 = 0                Answer: consistent system; three non-parallel planes intersect at the point  (-5, 1, 2)                          Video Solution

 

Example 2

7x + 3y + 9z = 20

5x + y + 7z = 12

x + 11y - 4z = 24                Answer: consistent system; three non-parallel planes intersect along the line r = (8, 0, -4) + t(-3, 1, 2), t E R             Video Solution

 

Example 3

x + y = 3

y + 2z = 7 

8x + 3y - 10z = 62                Answer: inconsistent system; three non-parallel planes form a triangular prism                             Video Solution

 

Example 4

x + 4y - 5z = 7

2x + 8y - 10z = 17

3x + 2y + 6z = 1                Answer: inconsistent system; two parallel non-coincident planes each intersected by a third plane                 Video Solution

 

Example 5

x + y - z = 4

2x - y + 5z = 11

6x - 3y + 15z = 33    Answer: consistent system; two coincident planes, third non-parallel plane intersect along line  r = (31/7 , 0, 3/7) + t(-4, 7, 3), t E R   Video

 

Example 6

x - 2y + z = 4

3x - 6y + 3z = 12

x/2 - y + z/2 = 2                Answer: consistent system; three coincident planes                             Video Solution

 

Example 7

4x - 3y + z = 10

8x - 6y + 2z = 20

x - 3/4 y + 1/4 z = 3                Answer: inconsistent system; two coincident planes and a third parallel, non-coincident plane                      Video Solution

 

Example 8

4x + 2y - z = 11

8x + 4y - 2z = 23

12x + 6y - 3z = 34                Answer: inconsistent system; three parallel non-coincident planes                             Video Solution