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Coplanar vectors

Definition. Vectors parallel to the same plane, or lie on the same plane are called coplanar vectors (Fig. 1).
Coplanar vectors
Fig. 1

It is always possible to find a plane parallel to the two random vectors, in that any two vectors are always coplanar.


Condition of vectors coplanarity



Examples of tasks

Example 1. Check whether the three vectors are coplanar a = {1; 2; 3}, b = {1; 1; 1}, c = {1; 2; 1}.

Solution: calculate a scalar triple product of vectors

a · [b × с] =    1     2     3    =
  1     1     1  
  1     2     1  

= 1·1·1 + 1·1·2 + 1·2·3 - 1·1·3 - 1·1·2 - 1·1·2 = 1 + 2 + 6 - 3 - 2 - 2 = 2

Answer: vectors are not coplanar as their scalar triple product is not zero.

Example 2. Prove that the three vectors a = {1; 1; 1}, b = {1; 3; 1} и c = {2; 2; 2} are coplanar.

Solution: calculate a scalar triple product of vectors

a · [b × с] =    1     1     1    =
  1     3     1  
  2     2     2  

= 1·2·3 + 1·1·2 + 1·1·2 - 1·2·3 - 1·1·2 - 1·1·2 = 6 + 2 + 2 - 6 - 2 - 2 = 0

Answer: vectors are coplanar as their scalar triple product is zero.

Example 3. Check whether the vectors are collinear a = {1; 1; 1}, b = {1; 2; 0}, c = {0; -1; 1}, d = {3; 3; 3}.

Solution: Find the number of linearly independent vectors, for this we write the values of the vectors in a matrix and run at her elementary transformations

(   1     1     1   )  ~
  1     2     0  
  0     -1     1  
  3     3     3  

from 2 row we subtract the 1-th row; from 4 row we subtract the 1-th row multiplied by 3;

(   1     1     1   )  ~  (   1     1     1   )  ~ 
  1 - 1     2 - 1     0 - 1     0     1     -1  
  0     -1     1     0     -1     1  
  3 - 3     3 - 3     3 - 3     0     0     0  

for 3 row add 2 row

(   1     1     1   )  ~  (   1     1     1   )
  0     1     -1     0     1     -1  
  0 + 0     -1 + 1     1 + (-1)     0     0     0  
  3 - 3     3 - 3     3 - 3     0     0     0  

Since there are two non-zero row, then among the given vectors only two linearly independent vectors.

Answer: vectors are coplanar since there only two linearly independent vectors.

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