Complanarity or coplanarity is a term from geometry - a sub-area of mathematics . Several points are called coplanar if they lie in one plane. Three vectors are considered to be coplanar if they are linearly dependent . One of the three vectors can thus be represented as a linear combination of the other two vectors; coplanar vectors lie in the same plane.
Complanarity study
To study the coplanarity of vectors one can Komplanaritätsuntersuchung be performed. Let three vectors be given . For the co-planarity, the equation must be satisfiable with , whereby 0 must not be simultaneously. The solution can be determined using a linear system of equations with n equations and the unknowns .
If the vectors come from a three-dimensional vector space, this check can be carried out with the late product : The vectors are coplanar if their late product is. It also applies that .
example
Three vectors and should be examined for co-planarity.
Approach :
With
The linear system of equations follows from the approach:
Substituting the result for r into equation (I) gives:
Equation (III) is fulfilled for and :
can be represented by a linear combination of and :
and it applies:
Thus , and are coplanar.
use
Complanarity studies are often carried out when determining the positional relationships between straight lines or straight lines and planes.