Did You Know?
The vector with magnitude equal to 1 is known as a unit vector.Notation
The notation for magnitude of a vector is two vertical bars.
 x 
To avoid confusion with absolute value, the magnitude X can also be written as;
 x 
However, the double bar notation is not used frequently.
Formula
The magnitude of a vector, v = (x,y), is given by the square root of squares of the endpoints x and y.
 v 
=
√  x^{2} + y^{2} 
Thus, if the two components (x, y) of the vector v is known, its magnitude can be calculated by Pythagoras theorem. If you have a vector (A,B) such that the components A and B are endpoints of the vector with coordinates A(x_{1}, y_{1}) and B(x_{2}, y_{2}), the vector magnitude is given by;
 v 
=
√  (x_{2}  x_{1}) ^{2} + (y_{2}  y_{1})^{2} 
Now let us take a look at a few examples that will help you understand the concept better.
Calculate magnitude of vector a = (4, 3).
Thus, the magnitude of vector a(4, 3) is 5 units.
 a 
=
√  4 ^{2} + 3^{2}  
√  16 + 9  
√  25  =  5 
Thus, the magnitude of vector a(4, 3) is 5 units.
Calculate magnitude of vector b = (3, 5).
Thus, the magnitude of vector b(3, 5) is 6 units.
 b 
=
√  (3) ^{2} + 5^{2}  
√  9 + 25  
√  36  =  6 
Thus, the magnitude of vector b(3, 5) is 6 units.
Vector Magnitude in Space
The aforementioned examples are for the vectors in 2D form. However, if you have to calculate vector magnitude in 3D space, you cannot use this formula. To calculate magnitude of u(x_{1}, x_{2}, x_{3}), the correct formula is;
 u  =  √

Similarly, if the coordinates of the endpoints A(x_{1}, y_{1}, z_{1}) and B(x_{2}, y_{2}, z_{2}) are mentioned, the formula is modified as;
 v 
=
√  (x_{2}  x_{1}) ^{2} + (y_{2}  y_{1})^{2} + (z_{2}  z_{1})^{2} 
Let us solve a few vectors in 3D space examples.
Calculate the magnitude of vector c = (2,5,3).
The magnitude of vector c is 6.16 units.
 c 
=
√  2^{2} + 5^{2} + 3^{2}  
√  4 + 25 + 9  
√  38  =  6.16 
The magnitude of vector c is 6.16 units.
Calculate the magnitude of vector u = (1,1,2).
Thus, the magnitude of the given vector is 2.44 units.
 u 
=
√  (1)^{2} + 1^{2} + 2^{2}  
√  1 + 1 + 4  
√  6  =  2.44 
Thus, the magnitude of the given vector is 2.44 units.
Calculate magnitude of vector with coordinates A(2,3,4) and B(1,3,2).
 u 
=
√  (12)^{2} + (33)^{2} + (24)^{2}  
√  9+ 0+ 36  
√  45  =  6.7 
Thus, 6.7 units is the magnitude of the given vector.