
Vector length. Vector magnitudePage Navigation:
Vector length  definitionDefinition. The length of the directed segment determines the numerical value of the vector is called the length of vector AB.The magnitude of a vector is the length of the vector. The length of the vector AB is denoted as AB. Basic relation. The length of vector a in Cartesian coordinates is the square root of the sum of the squares of its coordinates.
Vector length formulasVector length formula for twodimensional vectorIn the case of the plane problem the length of the vector a = {a_{x} ; a_{y}} can be found using the following formula: a = √a_{x}^{2} + a_{y}^{2} Vector length formula for threedimensional vectorIn the case of the spatial problem the length of the vector a = {a_{x} ; a_{y} ; a_{z}} can be found using the following formula: a = √a_{x}^{2} + a_{y}^{2} + a_{z}^{2} Vector length formula for arbitrary dimensions vectorIn the case of the n dimensional space problem the length of the vector a = {a_{1} ; a_{2}; ... ; a_{n}} can be found using the following formula:
Examples of tasksExamples of plane tasksExample 1. Find the length of the vector a = {2; 4}.
Solution: a = √2^{2} + 4^{2} = √4 + 16 = √20 = 2√5. Example 2. Find the length of the vector a = {3; 4}.
Solution: a = √3^{2} + (4)^{2} = √9 + 16 = √25 = 5. Examples of spatial tasksExample 3. Find the length of the vector a = {2; 4; 4}.
Solution: a = √2^{2} + 4^{2} + 4^{2} = √4 + 16 + 16 = √36 = 6. Example 4. Find the length of the vector a = {1; 0; 3}.
Solution: a = √(1)^{2} + 0^{2} + (3)^{2} = √1 + 0 + 9 = √10. Examples of n dimensional space tasksExample 5. Find the length of the vector a = {1; 3; 3; 1}.
Solution: a = √1^{2} + (3)^{2} + 3^{2} + (1)^{2} = √1 + 9 + 9 + 1 = √20 = 2√5 Example 6. Find the length of the vector a = {2; 4; 4; 6 ; 2}.
Solution: a = √2^{2} + 4^{2} + 4^{2} + 6^{2} + 2^{2} = √4 + 16 + 16 + 36 + 4 = √76 = 2√19.
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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