Complanarity

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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.