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Vector Coplanarity Calculator

This calculator checks if three vectors are coplanar and gives a detailed solution.




Vectors are coplanar if they lie in the same plane and are plotted from one point in three-dimensional space.


Specify the form of the first vector
Specify the form of the second vector
Specify the form of the third vector

Enter the coordinates of the first vector
a̅ = { ; ; }

Enter the coordinates of the second vector
b̅ = { ; ; }

Enter the coordinates of the third vector
c̅ = { ; ; }



How to Check if Two Vectors Are Coplanar

Example #1
Let's define coplanarity of three vectors of space. Vector coordinates are given by points.
Coordinates of point A of vector AB: (7; 9 ; -7)
Coordinates of point B of vector AB: (-1 ; 0 ; 16)
Coordinates of point C of vector CD: (-4 ; -6 ; 2)
Coordinates of point D of vector CD: (3 ; 0 ; 9)
Coordinates of point E of vector EF: (14 ; -1 ; 2)
Coordinates of point F of vector EF: (6 ; -3 ; -4)

Step by step solution:

Calculate the coordinates of the first vector from two points A and B:

AB = {xB - xA  ; yB - yA; zB - zA} = {-1 - 7 ; 0 - 9 ; 16 - 7} = {-8 ; -9 ; 9}

Calculate the coordinates of the second vector from two points C and D:

CD = {xD - xC  ; yD - yC; zD - zC} = {3 - (-4) ; 0 - (-6) ; 9 - 2} = {7 ; 6 ; 7}

Let's calculate the coordinates of the third vector from two points E and F:

EF = {xF - xE  ; yF - yE; zF - zE} = {6 - 14 ; -3 - (-1) ; -4 - 2} = {-8 ; -2 ; -6}

Three vectors are coplanar (lie in the same plane and are plotted from one point in space) if their mixed product is equal to zero. The mixed product of vectors is a number equal to the value of the determinant of the third-order matrix, the rows of which are the coordinates of the vectors being multiplied. Let's find the value of the determinant:
The formula for calculating the determinant for a third-order matrix:
AB โ‹… CD โ‹… EF =
ABxAByABz
CDxCDyCDz
EFxEFyEFz
= ABx
CDyCDz
EFyEFz
- ABy
CDxCDz
EFxEFz
+ ABz
CDxCDy
EFxEFy
= ABxCDyEFz - ABxCDzEFy - AByCDxEFz + AByCDzEFx + ABzCDxEFy - ABzCDyEFx
ABx = -8
ABy = -9
ABz = 9
CDx = 7
CDy = 6
CDz = 7
EFx = -8
EFy = -2
EFz = -6
AB โ‹… CD โ‹… EF =
(-8)(-9)9
767
(-8)(-2)(-6)
= (-8)
67
(-2)(-6)
- (-9)
77
(-8)(-6)
+ 9
76
(-8)(-2)
= (-8 โ‹… 6 โ‹… (-6)) - (-8 โ‹… 7 โ‹… (-2)) - (-9 โ‹… 7 โ‹… (-6)) + (-9 โ‹… 7 โ‹… (-8)) + (9 โ‹… 7 โ‹… (-2)) - (9 โ‹… 6 โ‹… (-8)) = 288 - 112 - 378 + 504 + (-126) - (-432) = 608
Vectors AB, CD, and EF are not coplanar since their mixed product is not zero.


Example #2
Let us define the coplanarity of the three vectors.
Vector coordinates a: (5 ; 1 ; 7)
Vector coordinates b: (2 ; 4 ; 6)
Vector coordinates c: (3 ; 8 ; 9)

Step by step solution:

Three vectors are coplanar (lie in the same plane and are plotted from one point in space) if their mixed product is equal to zero. The mixed product of vectors is a number equal to the value of the determinant of the third-order matrix, the rows of which are the coordinates of the vectors being multiplied. Let's find the value of the determinant:
The formula for calculating the determinant for a third-order matrix:
a โ‹… b โ‹… c =
axayaz
bxbybz
cxcycz
= ax
bybz
cycz
- ay
bxbz
cxcz
+ az
bxby
cxcy
= axbycz - axbzcy - aybxcz + aybzcx + azbxcy - azbycx
ax = 5
ay = 1
az = 7
bx = 2
by = 4
bz = 6
cx = 3
cy = 8
cz = 9
a โ‹… b โ‹… c =
517
246
389
= 5
46
89
- 1
26
39
+ 7
24
38
= (5 โ‹… 4 โ‹… 9) - (5 โ‹… 6 โ‹… 8) - (1 โ‹… 2 โ‹… 9) + (1 โ‹… 6 โ‹… 3) + (7 โ‹… 2 โ‹… 8) - (7 โ‹… 4 โ‹… 3) = 180 - 240 - 18 + 18 + 112 - 84 = -32
The vectors a, b, and c are not coplanar because their mixed product is not zero.
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