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

Basic relation.To find the 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(Ax ; Ay) and B(Bx ; By) can be found using the following formula

AB = {Bx - Ax ; By - Ay}

### Vector coordinates formula for spatial problems

In the case of the spatial problem the vector AB set by the coordinates of the points A(Ax ; Ay ; Az) and B(Bx ; By ; Bz) can be found using the following formula

AB = {Bx - Ax ; By - Ay ; Bz - Az}

### 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(A1 ; A2 ; ... ; An) and B(B1 ; B2 ; ... ; Bn) can be found using the following formula

AB = {B1 - A1 ; B2 - A2 ; ... ; Bn - An}

Example 1. Find the coordinates of vector AB, if A(1; 4), B(3; 1).

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

Example 2. Find the coordinates of point B of vector AB = {5; 1}, if coordinates of point A(3; -4).

Solution:

ABx = Bx - Ax   =>   Bx = ABx + Ax   =>   Bx = 5 + 3 = 8
ABy = By - Ay   =>   By = ABy + Ay   =>   By = 1 + (-4) = -3

Example 3. Find the coordinates of point A of vector AB = {5; 1}, if coordinates of point B(3; -4).

Solution:

ABx = Bx - Ax   =>   Ax = Bx - ABx   =>   Ax = 3 - 5 = -2
ABy = By - Ay   =>   Ay = By - ABy   =>   Ay = -4 - 1 = -5

Example 4. Find the coordinates of vector AB, if A(1; 4; 5), B(3; 1; 1).

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

Example 5. Find the coordinates of point B of vector AB = {5; 1; 2}, if coordinates of point A(3; -4; 3).

Solution:

ABx = Bx - Ax   =>   Bx = ABx + Ax   =>   Bx = 5 + 3 = 8
ABy = By - Ay   =>   By = ABy + Ay   =>   By = 1 + (-4) = -3
ABz = Bz - Az   =>   Bz = ABz + Az   =>   Bz = 2 + 3 = 5

Example 6. Find the coordinates of point A of vector AB = {5; 1; 4}, if coordinates of point B(3; -4; 1).

Solution:

ABx = Bx - Ax   =>   Ax = Bx - ABx   =>   Ax = 3 - 5 = -2
ABy = By - Ay   =>   Ay = By - ABy   =>   Ay = -4 - 1 = -5
ABz = Bz - Az   =>   Az = Bz - ABz   =>   Az = 1 - 4 = -3

### Examples of n dimensional space tasks

Example 7. Find the coordinates of vector AB, if A(1; 4; 5; 5; -3), B(3; 0; 1; -2; 5).

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

Example 8. Find the coordinates of point B of vector AB = {5; 1; 2; 1}, if coordinates of point A(3; -4; 3; 2).

Solution:

AB1 = B1 - A1   =>   B1 = AB1 + A1   =>   B1 = 5 + 3 = 8
AB2 = B2 - A2   =>   B2 = AB2 + A2   =>   B2 = 1 + (-4) = -3
AB3 = B3 - A3   =>   B3 = AB3 + A3   =>   B3 = 2 + 3 = 5
AB4 = B4 - A4   =>   B4 = AB4 + A4   =>   B4 = 1 + 2 = 3

Example 9. Find the coordinates of point A of vector AB = {5; 1; 4; 5}, if coordinates of point B(3; -4; 1; 8).

Solution:

AB1 = B1 - A1   =>   A1 = B1 - AB1   =>   A1 = 3 - 5 = -2
AB2 = B2 - A2   =>   A2 = B2 - AB2   =>   A2 = -4 - 1 = -5
AB3 = B3 - A3   =>   A3 = B3 - AB3   =>   A3 = 1 - 4 = -3
AB4 = B4 - A4   =>   A4 = B4 - AB4   =>   A4 = 8 - 5 = 3