Magnitude of A Vector: Formula, Examples and Questions

In JEE Main and IIT JAM, questions based on the calculation of vectors are often a part of a complex questions. It is, therefore, important to understand the concept instead of cramming the formula.

Point A, from where the vector AB starts, is called its initial point, and point B, where it ends, is called its terminal point. The distance between the initial and terminal points of a vector is called the magnitude of the vector (or length), denoted as |AB|.  The arrow indicates the direction of the vector. The magnitude of a vector is purely the value of the vector without taking into consideration the direction.  

$\Vert \mathbf{v}\Vert =\sqrt{{v_1}^2+{v_2}^2+\cdots +{v_n}^2}$

There are different formulae for calculating the magnitude of a vector:

$$1.\; Magnitude\; of\; vector\; with\; its\; components:\; |A|=\sqrt{{x_1}^2+{y_1}^2+{z_1}^2}$$

Say there is a vector called Ā = xi+ yĵ + zk̂. In this case, the magnitude of a vector will be calculated using the Pythagorean theorem. Any number of dimensions can be included in this. The magnitude will be the square root of sum of squares of all components. Hence, the magnitude of a vector will be $$\left|A\right|=\sqrt{x^2+y^2+z^2}$$

$$2.\; Magnitude\; of\; a\; Position\; Vector\; from\; Origin:\; |\stackrel{\to }{v}|=\sqrt{x^2+y^2}$$

Let us now suppose that you want to find the magnitude of a position vector from origin. Say, if any of the starting or the endpoint of a vector is at origin O(0,0) and the other point is B(x,y). In that case, the formula to calculate magnitude of a vector will be 

$$\left|\stackrel{\to }{B}\right|=\sqrt{x^2+y^2}$$

In 3D space, if the endpoint of vector is B(x,y,z), then the magnitude of a vector will be:

$$\left|\stackrel{\to }{B}\right|=\sqrt{x^2+y^2+z^2}$$

$$3.\; Magnitude\; of\; a\; Vector\; Formula\; Between\; Two\; Points:\; |\stackrel{\to }{v}|=\sqrt{{\left({\mathrm{((x}}_2-x_{\mathrm{1)}}\right)}^2+{\left({\mathrm{(y}}_2-y_{\mathrm{1))}}\right)}^2}$$

Suppose, the starting point of vector is (x₁, y₁) and its endpoint is (x₂, y₂). In this case, the magnitude of a vector  $\stackrel{\to }{AB}$ will be

You will need to follow the below-given steps in order to calculate the magnitude of a vector:

• First, you need to identify the components of a vector.
• Calculate the sum of squares of all components
• Once you have found the sum, take the square root of that sum. 

Let us understand this with an example to calculate the magnitude of a 3d vector. The vector is  $\mathbf{v}=\left(3,,,-,4,,,12\right)$.

The magnitude of a vector will be $\parallel \mathbf{v}\parallel =13$