
Addition and subtraction of vectorsPage Navigation:
Definition. Addition of vectors (sum of vectors) a + b is operation of a finding of a vector c , coordinates of which are equal to the sum of all corresponding coordinates of vectors a and b , can be defined as:
c_{i} = a_{i} + b_{i} Definition. Subtraction of vectors (difference of vectors) a  b is operation of a finding of a vector c , coordinates of which are equal to a difference of all corresponding coordinates of vectors a and b , can be defined as:
c_{i} = a_{i}  b_{i} Addition and subtraction of vectors  formulasThe formulas of addition and subtraction of vectors for plane problemsIn 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 problemsIn 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 problemsIn the case of the n dimensional space problem the sum and difference of vectors a = {a_{1} ; a_{2} ; ... ; a_{n}} and b = {b_{1} ; b_{2} ; ... ; b_{n}} can be found using the following formulas: a + b = {a_{1} + b_{1}; a_{2} + b_{2}; ... ; a_{n} + b_{n}} a  b = {a_{1}  b_{1}; a_{2}  b_{2}; ... ; a_{n}  b_{n}} Addition and subtraction of vectors  examplesExamples of plane tasksExample 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 tasksExample 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 tasksExample 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
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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