# Vector length. Vector magnitude

## Vector length - definition

**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|.

**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 = {a_{x} ; a_{y}} can be found using the following formula:

_{x}

^{2}+ a

_{y}

^{2}

### Vector length formula for three-dimensional vector

In 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:

_{x}

^{2}+ a

_{y}

^{2}+ a

_{z}

^{2}

### Vector length formula for arbitrary dimensions vector

In 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:

|a| = ( | n | a_{i}^{2})^{1/2} |

Σ | ||

i=1 |

## Examples of tasks

### Examples of plane tasks

**Solution:** |a| = √2^{2} + 4^{2} = √4 + 16 = √20 = 2√5.

**Solution:** |a| = √3^{2} + (-4)^{2} = √9 + 16 = √25 = 5.

### Examples of spatial tasks

**Solution:** |a| = √2^{2} + 4^{2} + 4^{2} = √4 + 16 + 16 = √36 = 6.

**Solution:** |a| = √(-1)^{2} + 0^{2} + (-3)^{2} = √1 + 0 + 9 = √10.

### Examples of n dimensional space tasks

**Solution:** |a| = √1^{2} + (-3)^{2} + 3^{2} + (-1)^{2} = √1 + 9 + 9 + 1 = √20 = 2√5

**Solution:** |a| = √2^{2} + 4^{2} + 4^{2} + 6^{2} + 2^{2} = √4 + 16 + 16 + 36 + 4 = √76 = 2√19.

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