Magnitude of a Vector Formula

The magnitude of a vector formula is used to calculate the length of a vector and is denoted by |v|. The magnitude of a vector is always a positive number or zero it cannot be a negative number. Learn more about vectors here and know everything about it including their representations, operations, and characteristics.

Formula of Magnitude of a Vector

The magnitude of a vector can be calculated in two scenarios. In one case, the magnitude is calculated for a vector when its endpoint is at origin (0,0) while in the other case, the starting and ending point of the vector is at certain points (x1, y1) and (x2, y2) respectively. The formulas to claculate the length for each of the cases is given below.

The Magnitude of a Vector Formulas
Magnitude Formula for a Vector When End Point is Origin \(\left|v\right|=\sqrt{x^{2}+y^{2}}\)
Magnitude Formula for a Vector when starting points are (x1, y1) and endpoints are (x2, y2) \(\left|v\right|=\sqrt{\left(x_{2}+x_{1}\right)^{2}+\left(y_{2}+y_{1}\right)^{2}}\)

Solved Examples Using The Vector Magnitude Formula

Question: Find the magnitude of the vector with \(\vec{u}\) = (3,5) ?

Solution: 

Given, \(\vec{u}\) = (3,5)

Use magnitude formula,

|v| = √(x2 + y2)

\(|v|= \sqrt{3^{2}+5^{2}}\) \(|v| = \sqrt{9 + 25}\)

|v|= 5.83


Practise This Question

The lines 2x - 3y = 5 and 3x - 4y = 7 are the diameters of a circle of area 154 square units. The equation of the circle is