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 .
![\ alpha {\ vec a} + \ beta {\ vec b} + \ gamma {\ vec c} = {\ vec 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1eb691b12e67d7af1d9d038ab7abf0ed6220136)
![\ alpha, \ beta, \ gamma \ in \ mathbb {R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfa78c197a825c131b114e4dea7a20c94e2c7897)
![\ alpha, \ beta, \ gamma](https://wikimedia.org/api/rest_v1/media/math/render/svg/301cc1b37ba8f0fb0c9bedee5efa5e0b5bc9e791)
![\ alpha, \ beta, \ gamma](https://wikimedia.org/api/rest_v1/media/math/render/svg/301cc1b37ba8f0fb0c9bedee5efa5e0b5bc9e791)
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 .
![\ vec a, \ vec b, \ vec c](https://wikimedia.org/api/rest_v1/media/math/render/svg/befa941df6751fd0d377ccd7d31130ef372a3b8c)
![{\ displaystyle ({\ vec {a}}, {\ vec {b}}, {\ vec {c}}) = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d508d1f9e0cc13febaa97c80d7018d39afdacf69)
![\ det ({\ vec a}, {\ vec b}, {\ vec c}) = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/0560efd682cc64500603de6cb2ad3381bd5792d5)
example
Three vectors and should be examined for co-planarity.
![{\ vec a} = {\ begin {pmatrix} 2 \\ 4 \\ 6 \ end {pmatrix}}, {\ vec b} = {\ begin {pmatrix} 2 \\ 6 \\ 7 \ end {pmatrix} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/add7c721694ad8a45377e23f56008d69c5f6bc6d)
![{\ vec c} = {\ begin {pmatrix} 2 \\ 0 \\ 4 \ end {pmatrix}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e8b5fcd144d94d408b16d5967f2dc5e9f199f7f)
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 :
![r = {\ frac {2} {3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10f838834f5c65bbb0a87716fbab4cfb159e095a)
![s = {\ frac {1} {3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/207a596363f3d4970bdfa6912ab0db48c09c1ac1)
can be represented by a linear combination of and :
![{\ vec {b}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216)
![\ vec c](https://wikimedia.org/api/rest_v1/media/math/render/svg/965bd8710781b710cbfdb79da0b4e3b097bef506)
and it applies:
Thus , and are coplanar.
![{\ vec {a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16)
![{\ vec {b}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216)
![\ vec c](https://wikimedia.org/api/rest_v1/media/math/render/svg/965bd8710781b710cbfdb79da0b4e3b097bef506)
use
Complanarity studies are often carried out when determining the positional relationships between straight lines or straight lines and planes.