Coplanar vectors

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.