Study of mathematics online.
Study math with us and make sure that "Mathematics is easy!"

# Addition and subtraction of vectors

Definition.
Addition of vectors (sum of vectors)
a
+
b
is operation of a finding of a vector
c
, coordinates of which are equal to the sum of all corresponding coordinates of vectors
a
and
b
, can be defined as:

ci = ai + bi

Definition.
Subtraction of vectors (difference of vectors)
a
-
b
is operation of a finding of a vector
c
, coordinates of which are equal to a difference of all corresponding coordinates of vectors
a
and
b
, can be defined as:

ci = ai - bi

## Addition and subtraction of vectors - formulas

### The formulas of addition and subtraction of vectors for plane problems

In the case of the plane problem the sum and difference of vectors a = {ax ; ay} and b = {bx ; by} can be found using the following formulas:

a + b = {ax + bx; ay + by}

a - b = {ax - bx; ay - by}

### The formulas of addition and subtraction of vectors for spatial problems

In the case of the spatial problem the sum and difference of vectors a = {ax ; ay ; az} and b = {bx ; by ; bz} can be found using the following formulas:

a + b = {ax + bx; ay + by; az + bz}

a - b = {ax - bx; ay - by; az - bz}

### The formulas of addition and subtraction of vectors for n dimensional space problems

In the case of the n dimensional space problem the sum and difference of vectors a = {a1 ; a2 ; ... ; an} and b = {b1 ; b2 ; ... ; bn} can be found using the following formulas:

a + b = {a1 + b1; a2 + b2; ... ; an + bn}

a - b = {a1 - b1; a2 - b2; ... ; an - bn}

## Addition and subtraction of vectors - examples

### Examples of plane tasks

Example 1. Find the sum of vectors a = {1; 2} and b = {4; 8}.

Solution:

a + b = {1 + 4; 2 + 8} = {5; 10}
Example 2. Find the difference of vectors a = {1; 2} and b = {4; 8}.

Solution:

a - b = {1 - 4; 2 - 8} = {-3; -6}

### Examples of spatial tasks

Example 3. Find the sum of vectors a = {1; 2; 5} and b = {4; 8; 1}.

Solution:

a + b = {1 + 4; 2 + 8; 5 + 1} = {5; 10; 6}
Example 4. Find the difference of vectors a = {1; 2; 5} and b = {4; 8; 1}.

Solution:

a - b = {1 - 4; 2 - 8; 5 - 1} = {-3; -6; 4}

### Examples of n dimensional space tasks

Example 5. Find the sum of vectors a = {1; 2; 5; 9} and b = {4; 8; 1; -20}.

Solution:

a + b = {1 + 4; 2 + 8; 5 + 1; 9 + (-20)} = {5; 10; 6; -11}
Example 6. Find the difference of vectors a = {1; 2; 5; -1; 5} and b = {4; 8; 1; -1; 2}.

Solution:

a - b = {1 - 4; 2 - 8; 5 - 1; -1 - (-1); 5 - 2} = {-3; -6; 4; 0; 3}