
Component form of a vector with initial point and terminal pointPage Navigation:
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 pointsVector coordinates formula for plane problemsIn 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 AB = {B_{x}  A_{x} ; B_{y}  A_{y}}
Vector coordinates formula for spatial problemsIn 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 AB = {B_{x}  A_{x} ; B_{y}  A_{y} ; B_{z}  A_{z}}
Vector coordinates formula for n dimensional space problemsIn 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 AB = {B_{1}  A_{1} ; B_{2}  A_{2} ; ... ; B_{n}  A_{n}}
Examples of tasksExamples of plane tasksExample 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: AB_{x} = B_{x}  A_{x} => B_{x} = AB_{x} + A_{x} => B_{x} = 5 + 3 = 8AB_{y} = B_{y}  A_{y} => B_{y} = AB_{y} + A_{y} => B_{y} = 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: AB_{x} = B_{x}  A_{x} => A_{x} = B_{x}  AB_{x} => A_{x} = 3  5 = 2AB_{y} = B_{y}  A_{y} => A_{y} = B_{y}  AB_{y} => A_{y} = 4  1 = 5 Answer: A(2; 5). Examples of spatial tasksExample 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: AB_{x} = B_{x}  A_{x} => B_{x} = AB_{x} + A_{x} => B_{x} = 5 + 3 = 8AB_{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). Example 6. Find the coordinates of point A of vector AB = {5; 1; 4}, if coordinates of point B(3; 4; 1).
Solution: AB_{x} = B_{x}  A_{x} => A_{x} = B_{x}  AB_{x} => A_{x} = 3  5 = 2AB_{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 tasksExample 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: AB_{1} = B_{1}  A_{1} => B_{1} = AB_{1} + A_{1} => B_{1} = 5 + 3 = 8AB_{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). 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: AB_{1} = B_{1}  A_{1} => A_{1} = B_{1}  AB_{1} => A_{1} = 3  5 = 2AB_{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).
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
Scalarvector 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
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