Addition and subtraction of vectors

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Addition of vectors - definition
Subtraction of vectors - definition
Addition and subtraction of vectors - formulas
Addition and subtraction of vectors - examples

Online calculator. Addition and subtraction of two vectors.

Definition.

Addition of vectors (sum of vectors)

is operation of a finding of a vector

, coordinates of which are equal to the sum of all corresponding coordinates of vectors

and

, can be defined as:

c_i = a_i + b_i

Definition.

Subtraction of vectors (difference of vectors)

is operation of a finding of a vector

, coordinates of which are equal to a difference of all corresponding coordinates of vectors

and

, can be defined as:

c_i = a_i - b_i

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 = {a_x ; a_y} and b = {b_x ; b_y} can be found using the following formulas:

a + b = {a_x + b_x; a_y + b_y}

a - b = {a_x - b_x; a_y - b_y}

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 = {a_x ; a_y ; a_z} and b = {b_x ; b_y ; b_z} can be found using the following formulas:

a + b = {a_x + b_x; a_y + b_y; a_z + b_z}

a - b = {a_x - b_x; a_y - b_y; a_z - b_z}

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 ; ... ; a_n} and b = {b₁ ; b₂ ; ... ; b_n} can be found using the following formulas:

a + b = {a₁ + b₁; a₂ + b₂; ... ; a_n + b_n}

a - b = {a₁ - b₁; a₂ - b₂; ... ; a_n - b_n}

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}

Vectors Vectors Definition. Main information Component form of a vector with initial point and terminal point Length of a vector Direction cosines of a vector Equal vectors Orthogonal vectors Collinear vectors Coplanar vectors Angle between two vectors Vector projection Addition and subtraction of vectors Scalar-vector multiplication Dot product of two vectors Cross product of two vectors (vector product) Scalar triple product (mixed product) Linearly dependent and linearly independent vectors Decomposition of the vector in the basis

Online calculators with vectors

Tasks and exercises with vector 2D

Tasks and exercises with vector 3D

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