Coplanarity

Coplanarity or coplanarity is a term used in geometry - a branch of mathematics. Several points are called coplanar if they lie in a plane. Three vectors are considered to be coplanar if they are linearly dependent. One of the three vectors thus be represented as a linear combination of the other two vectors; coplanar vectors lie in the same plane.

Komplanaritätsuntersuchung

For checking the coplanarity of vectors can be performed Komplanaritätsuntersuchung. Given three vectors. For the coplanarity equation must be fulfilled, which must not be 0 simultaneously. The solution can be determined by means of a linear system with n equations and unknowns.

Derive the vectors a three-dimensional vector space, so can perform with the scalar triple this exam. The vectors are coplanar if.

Example

Three vectors and should be examined for coplanarity.

Approach:

With

The recognition of the linear system of equations follows:

Substituting the result for r in equation (I ) gives:

Equation ( III) and is satisfied:

Is determined by a linear combination of and be represented:

And we have:

Thus, and coplanar.

Use

Komplanaritätsuntersuchungen are often performed in determining the positional relationships between straight lines or straight lines and planes.

• Euclidean geometry