# Component form of a vector with initial point and terminal point

**coordinates of the vector**AB, knowing the coordinates of its initial point A and terminal point B is necessary subtract the appropriate coordinates of initial point from terminal point.

## Formulas determining coordinates of a vector by given coordinates of its initial and terminal points

### Vector coordinates formula for plane problems

In the case of the plane problem the vector AB set by the coordinates of the points A(A_{x} ; A_{y}) and B(B_{x} ; B_{y}) can be found using the following formula

_{x}- A

_{x}; B

_{y}- A

_{y}}

### Vector coordinates formula for spatial problems

In the case of the spatial problem the vector AB set by the coordinates of the points A(A_{x} ; A_{y} ; A_{z}) and B(B_{x} ; B_{y} ; B_{z}) can be found using the following formula

_{x}- A

_{x}; B

_{y}- A

_{y}; B

_{z}- A

_{z}}

### Vector coordinates formula for n dimensional space problems

In the case of the n dimensional space problem the vector AB set by the coordinates of the points A(A_{1} ; A_{2} ; ... ; A_{n}) and B(B_{1} ; B_{2} ; ... ; B_{n}) can be found using the following formula

_{1}- A

_{1}; B

_{2}- A

_{2}; ... ; B

_{n}- A

_{n}}

## Examples of tasks

### Examples of plane tasks

**Solution:** AB = {3 - 1; 1 - 4} = {2; -3}.

**Solution:**

_{x}= B

_{x}- A

_{x}=> B

_{x}= AB

_{x}+ A

_{x}=> B

_{x}= 5 + 3 = 8

AB

_{y}= B

_{y}- A

_{y}=> B

_{y}= AB

_{y}+ A

_{y}=> B

_{y}= 1 + (-4) = -3

**Answer:** B(8; -3).

**Solution:**

_{x}= B

_{x}- A

_{x}=> A

_{x}= B

_{x}- AB

_{x}=> A

_{x}= 3 - 5 = -2

AB

_{y}= B

_{y}- A

_{y}=> A

_{y}= B

_{y}- AB

_{y}=> A

_{y}= -4 - 1 = -5

**Answer:** A(-2; -5).

### Examples of spatial tasks

**Solution:** AB = {3 - 1; 1 - 4; 1 - 5} = {2; -3; -4}.

**Solution:**

_{x}= B

_{x}- A

_{x}=> B

_{x}= AB

_{x}+ A

_{x}=> B

_{x}= 5 + 3 = 8

AB

_{y}= B

_{y}- A

_{y}=> B

_{y}= AB

_{y}+ A

_{y}=> B

_{y}= 1 + (-4) = -3

AB

_{z}= B

_{z}- A

_{z}=> B

_{z}= AB

_{z}+ A

_{z}=> B

_{z}= 2 + 3 = 5

**Answer:** B(8; -3; 5).

**Solution:**

_{x}= B

_{x}- A

_{x}=> A

_{x}= B

_{x}- AB

_{x}=> A

_{x}= 3 - 5 = -2

AB

_{y}= B

_{y}- A

_{y}=> A

_{y}= B

_{y}- AB

_{y}=> A

_{y}= -4 - 1 = -5

AB

_{z}= B

_{z}- A

_{z}=> A

_{z}= B

_{z}- AB

_{z}=> A

_{z}= 1 - 4 = -3

**Answer:** A(-2; -5; -3).

### Examples of n dimensional space tasks

**Solution:** AB = {3 - 1; 0 - 4; 1 - 5; -2 - 5; 5 - (-3)} = {2; -4; -4; -7; 8}.

**Solution:**

_{1}= B

_{1}- A

_{1}=> B

_{1}= AB

_{1}+ A

_{1}=> B

_{1}= 5 + 3 = 8

AB

_{2}= B

_{2}- A

_{2}=> B

_{2}= AB

_{2}+ A

_{2}=> B

_{2}= 1 + (-4) = -3

AB

_{3}= B

_{3}- A

_{3}=> B

_{3}= AB

_{3}+ A

_{3}=> B

_{3}= 2 + 3 = 5

AB

_{4}= B

_{4}- A

_{4}=> B

_{4}= AB

_{4}+ A

_{4}=> B

_{4}= 1 + 2 = 3

**Answer:** B(8; -3; 5; 3).

**Solution:**

_{1}= B

_{1}- A

_{1}=> A

_{1}= B

_{1}- AB

_{1}=> A

_{1}= 3 - 5 = -2

AB

_{2}= B

_{2}- A

_{2}=> A

_{2}= B

_{2}- AB

_{2}=> A

_{2}= -4 - 1 = -5

AB

_{3}= B

_{3}- A

_{3}=> A

_{3}= B

_{3}- AB

_{3}=> A

_{3}= 1 - 4 = -3

AB

_{4}= B

_{4}- A

_{4}=> A

_{4}= B

_{4}- AB

_{4}=> A

_{4}= 8 - 5 = 3

**Answer:** A(-2; -5; -3; 3).

*Add the comment*