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# Vector length. Vector magnitude

## Vector length - definition

Definition.
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 formulas

### Vector length formula for two-dimensional vector

In the case of the plane problem the length of the vector a = {ax ; ay} can be found using the following formula:

|a| = √ax2 + ay2

### Vector length formula for three-dimensional vector

In the case of the spatial problem the length of the vector a = {ax ; ay ; az} can be found using the following formula:

|a| = √ax2 + ay2 + az2

### Vector length formula for arbitrary dimensions vector

In the case of the n dimensional space problem the length of the vector a = {a1 ; a2; ... ; an} can be found using the following formula:

 |a| = ( n ai2)1/2 Σ i=1

Example 1. Find the length of the vector a = {2; 4}.

Solution: |a| = √22 + 42 = √4 + 16 = √20 = 2√5.

Example 2. Find the length of the vector a = {3; -4}.

Solution: |a| = √32 + (-4)2 = √9 + 16 = √25 = 5.

Example 3. Find the length of the vector a = {2; 4; 4}.

Solution: |a| = √22 + 42 + 42 = √4 + 16 + 16 = √36 = 6.

Example 4. Find the length of the vector a = {-1; 0; -3}.

Solution: |a| = √(-1)2 + 02 + (-3)2 = √1 + 0 + 9 = √10.

### Examples of n dimensional space tasks

Example 5. Find the length of the vector a = {1; -3; 3; -1}.

Solution: |a| = √12 + (-3)2 + 32 + (-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| = √22 + 42 + 42 + 62 + 22 = √4 + 16 + 16 + 36 + 4 = √76 = 2√19.