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# Vector Dot Product Calculator. By Using Vector Coordinates.

## This calculator calculates the dot (scalar) product of vectors using coordinates and gives a detailed solution to all stages of the calculation.

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 = { ; }

Specify coordinates of vector b
b̅ = { ; }

How to Find the Dot Product of Vectors in 2d and 3d Space

Example #1
Let's find the dot product of vectors in two-dimensional space. The coordinates of both vectors are given by points.
Point A coordinates of vector AB: (5 ; 9)
Point B coordinates of vector AB: (-2 ; 11)
Point C coordinates of vector CD: (0 ; 12)
Point D coordinates of vector CD: (-3 ; 1)

Step by step solution:

Let's calculate the coordinates of the first vector from two points A and B:

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

Let's calculate the coordinates of the second vector from two points C and D:

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

Find the dot product of vectors: AB and CD

AB โ CD = ABxCDx + AByCDy = -7 โ (-3) + 2 โ (-11) = 21 + (-22) = -1

Example #2
Find the dot product of the vectors in two-dimensional space.
The coordinates of the vector a: (5 ; 9)
The coordinates of the vector b: (-1 ; 7)

Step by step solution:

Find the dot product of vectors: a and b

a โ b = axbx + ayby = 5 โ (-1) + 9 โ 7 = -5 + 63 = 58

Example #3
Let's find the dot product of vectors 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:

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}

Find the dot product of vectors: AB and CD

AB โ CD = ABxCDx + AByCDy + ABzCDz = -8 โ 7 + (-1/5) โ 6 + (-275/4) โ 7 = -56 + (-6/5) + (-1925/4) = -10769/20 = -538.45

Example #4
Let's find the dot 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:

Find the dot product of vectors: a and b

a โ b = axbx + ayby + azbz = 5 โ 2 + 1 โ 4 + 7 โ 6 = 10 + 4 + 42 = 56
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