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# Vector Addition and Subtraction Calculator

## This calculator calculates the sum and difference of vectors 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 = { ; }

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

How to Add or Subtract Two Vectors

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

Step by step solution:

In order to add two vectors, add their coordinates. The result of vector addition AB and CD will be a vector c

Let's calculate the coordinates of the first vector of a given point A and B:

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

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

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

c = AB + CD = {ABx + CDx ; ABy + CDy} = {-7 + (-3) ; 2 + (-11)} = {-10 ; -9}

Example # 2
Let's find the difference of vectors in two-dimensional space.
Vector coordinates a : (5; 9)
Vector coordinates b : (-1; 7)

Step by step solution:

In order to subtract two vectors, you need to subtract their coordinates. By subtracting the vectors a and b there will be a vector c
c = a - b = {ax - bx ; ay - by} = {5 - (-1) ; 9 - 7} = {6 ; 2}
Example # 3
Let's find the difference 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)
The coordinates of point B of the vector AB : (-1; 0; 2/8)
Point C coordinates of the vector CD : (-4; -6; 2)
Point D coordinates of the vector CD : (3; 0; 9)

Step by step solution:

In order to subtract two vectors, you need to subtract their coordinates. The result of subtracting vectors AB and CD there will be a vector c

Let's calculate the coordinates of the first vector by 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 by two points C and D:

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

c = AB - CD = {ABx - CDx ; ABy - CDy ; ABz - CDz} = {-8 - 7 ; -1/5 - 6 ; -275/4 - 7} = {-15 ; -31/5 ; -303/4}= {-15 ; -6.2 ; -75.75}

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

Step by step solution:

In order to add two vectors, you need to add their coordinates. The result of the addition of vectors a and b will be a vector c
c = a + b = {ax + bx ; ay + by ; az + bz} = {5 + 2 ; 1 + 4 ; 7 + 6} = {7 ; 5 ; 13}
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