Vector length. Vector magnitude

The length of the directed segment determines the numerical value of the vector is called the length of vector 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 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:

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:

Vector length formula for arbitrary dimensions vector

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

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.

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.