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# Vector Product Calculator. By Using Vector Coordinates.

## This calculator calculates the cross product of vectors using their coordinates and gives a detailed solution to all stages of the calculation.

The product of two vectors is a vector perpendicular to both initial vectors, the length of which is equal to the area of the parallelogram formed by the initial vectors.The cross product of two vectors is defined only in three-dimensional 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 Calculate the Cross Products of Vectors

Example #1
Let's find the cross product in three-dimensional space. 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:

NÌ… = [AB Ã— CD] =
 iÌ… jÌ… kÌ… ABx ABy ABz CDx CDy CDz
= iÌ…
 ABy ABz CDy CDz
- jÌ…
 ABx ABz CDx CDz
+ kÌ…
 ABx ABy CDx CDy
= iÌ… AByCDz - iÌ… ABzCDy - jÌ… ABxCDz + jÌ… ABzCDx + kÌ… ABxCDy - kÌ… AByCDx =
iÌ… (AByCDz - CDyABz) - jÌ… (ABxCDz - CDxABz) + kÌ… (ABxCDy - CDxABy)

Let's 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}

Let's 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}

ABx = -8
ABy = -1/5
ABz = -275/4
CDx = 7
CDy = 6
CDz = 7

NÌ… = [AB Ã— CD] = iÌ… (-1/5 â‹… 7 - 6 â‹… (-275/4)) - jÌ… (-8 â‹… 7 - 7 â‹… (-275/4)) + kÌ… (-8 â‹… 6 - 7 â‹… (-1/5)) = iÌ… (-7/5 - (-825/2)) - jÌ… (-56 - (-1925/4)) + kÌ… (-48 - (-7/5)) = (4111/10)iÌ… - (1701/4)jÌ… - (233/5)kÌ…
NÌ… = [AB Ã— CD] = (4111/10)iÌ… - (1701/4)jÌ… - (233/5)kÌ…
NÌ… = {4111/10 ; -1701/4 ; -233/5}
NÌ… = {411.1 ; -425.25 ; -46.6}

Example #2
Let's find the vector product of vectors in three-dimensional space.
The coordinates of the vector a: (5 ; 1 ; 7)
The coordinates of the vector b: (2 ; 4 ; 6)

Step by step solution:

NÌ… = [a Ã— b] =
 iÌ… jÌ… kÌ… ax ay az bx by bz
= iÌ…
 ay az by bz
- jÌ…
 ax az bx bz
+ kÌ…
 ax ay bx by
= iÌ… aybz - iÌ… azby - jÌ… axbz + jÌ… azbx + kÌ… axby - kÌ… aybx =
iÌ… (aybz - byaz) - jÌ… (axbz - bxaz) + kÌ… (axby - bxay)
ax = 5
ay = 1
az = 7
bx = 2
by = 4
bz = 6

NÌ… = [a Ã— b] = iÌ… (1 â‹… 6 - 4 â‹… 7) - jÌ… (5 â‹… 6 - 2 â‹… 7) + kÌ… (5 â‹… 4 - 2 â‹… 1) = iÌ… (6 - 28) - jÌ… (30 - 14) + kÌ… (20 - 2) = -22iÌ… -16jÌ… +18kÌ…
NÌ… = [a Ã— b] = -22iÌ… -16jÌ… +18kÌ…
NÌ… = {-22 ; -16 ; 18}
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