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Vector Collinearity Check Calculator

This calculator checks if two vectors are collinear (parallel) and gives a detailed solution.




Collinearity of vectors is the ratio of parallelism of vectors, so two nonzero vectors are collinear (parallel) if they lie on parallel lines or on one straight line.

Dimension of space
Specify the form of the first vector
Specify the form of the second vector

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


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



How to Check if Two Vectors Are Collinear (Parallel)

Example #1
Determine whether two vectors located in two-dimensional space are collinear (parallel). The coordinates of both vectors are given by points.
Coordinates of point A of vector AB: (5 ; 9)
Coordinates of point B of vector AB: (-2 ; 11)
Coordinates of point C of vector CD: (0 ; 12)
Coordinates of point D of vector CD: (-3 ; 1)

Step by step solution:

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

AB = {xB - xA  ; yB - yA} = {-2 - 5 ; 11 - 9} = {-7 ; 2}

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

CD = {xD - xC  ; yD - yC} = {-3 - 0 ; 1 - 12} = {-3 ; -11}

Let us check if the vectors AB and CD have a coefficient of proportionality q (which has the same value for all equations in the system), for this we compose the system of equations and check whether the equality holds
ABx = q โ‹… CDx
ABy = q โ‹… CDy

-7 = q โ‹… (-3) then, q = -7
-3
= 7
3

2 = q โ‹… (-11) then, q = 2
-11
= -2
11

7
3
โ‰  -2
11

In the first equation, the coefficient of proportionality is 7
3
, in the second -2
11

Therefore, the system is incompatible and has no solutions. The coordinates of vectors AB and CD are not proportional, which means that vectors AB and CD are not collinear.


Example #2
Determine whether two vectors located in two-dimensional space are collinear (parallel).
Vector coordinates a: (2 ; 6)
Vector coordinates b: (7 ; 21)

Step by step solution:

Let us check if the vectors a and b have a coefficient of proportionality q (which has the same value for all equations in the system), for this we compose the system of equations and check whether the equality holds
ax = q โ‹… bx
ay = q โ‹… by

2 = q โ‹… 7 then, q = 2
7

6 = q โ‹… 21 then, q = 6
21
= 2
7

2
7
= 2
7

In the first equation, the coefficient of proportionality is 2
7
, in the second 2
7
The coefficient of proportionality in each equation has the same value, therefore the coordinates of the vectors a and b are proportional and therefore the vectors are collinear. The value of the coefficient of proportionality is greater than zero, which means that vectors a and b are co-directed.


Example #3
Determine whether two vectors in three-dimensional space are collinear (parallel). The coordinates of both vectors are given by points.
Coordinates of point A of vector AB: (7; 0.2 ; 69)
Coordinates of point B of vector AB: (-1 ; 0 ; 2/8)
Coordinates of point C of vector CD: (-4 ; -6 ; 2)
Coordinates of point D of vector CD: (3 ; 0 ; 9)

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 - 0.2 ; 2/8 - 69} = {-8 ; 1/5 ; -275/4}

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 us check if the vectors AB and CD have a coefficient of proportionality q (which has the same value for all equations in the system), for this we compose the system of equations and check whether the equality holds
ABx = q โ‹… CDx
ABy = q โ‹… CDy
ABz = q โ‹… CDz

-8 = q โ‹… 7 then, q = -8
7

1/5 = q โ‹… 6 then, q = 1/5
6
= 1
30

-275/4 = q โ‹… 7 then, q = -275/4
7
= -275
28

-8
7
โ‰  1
30
โ‰  -275
28

In the first equation, the coefficient of proportionality is -8
7
, in the second 1
30
and in the third -275
28

Therefore, the system is incompatible and has no solutions. The coordinates of vectors AB and CD are not proportional, which means that vectors AB and CD are not collinear.


Example #4
Determine whether two vectors of vectors located in three-dimensional space are collinear (parallel).
Vector coordinates a: (0 ; 1 ; 7)
Vector coordinates b: (2 ; 0 ; 6)

Step by step solution:

Let us check if the vectors a and b have a coefficient of proportionality q (which has the same value for all equations in the system), for this we compose the system of equations and check whether the equality holds
ax = q โ‹… bx
ay = q โ‹… by
az = q โ‹… bz

0 = q โ‹… 2 then, q = 0
2
= 0

1 = q โ‹… 0

7 = q โ‹… 0

Let's check, substitute the value of the coefficient in the equations

0 =
0
โ‹… 2

1 โ‰ 
0
โ‹… 0

7 โ‰ 
0
โ‹… 0
Therefore, the system is incompatible and has no solutions. The coordinates of vectors a and b are not proportional, which means that vectors a and b are not collinear.
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