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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.
vector

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}

Examples of tasks

Examples of plane tasks

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

Answer: B(8; -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

Answer: A(-2; -5).

Examples of spatial tasks

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

Answer: B(8; -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

Answer: A(-2; -5; -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

Answer: B(8; -3; 5; 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

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

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