Magnitude of a Vector Formula
The magnitude of a vector formula helps to summarize the numeric value for a given vector. A vector has a direction and a magnitude. The individual measures of the vector along the xaxis, yaxis, and zaxis are summarized using this magnitude of a vector formula. It is denoted by \(\overrightarrow{ \text v}\). The magnitude of a vector is always a positive number or zero, i.e., it cannot be a negative number. Let us understand the magnitude of a vector formula using a few solved examples in the end.
What Is the Magnitude of a Vector Formula?
The magnitude of a vector \(\overrightarrow{ \text A}\) is A. For a given vector with direction ratios along the xaxis, yaxis, and zaxis, the magnitude of the vector is equal to the square root of the sum of the square of its direction ratios. This can be clearly understood from the below magnitude of a vector formula.
Magnitude of a Vector Formula
 For a vector \(\bar A = x \hat i + y \hat j + z \hat k \) its magnitude is:
\( A = \sqrt{x_1^2 + y_1^2 + z_1^2}\)
 The magnitude is calculated for a vector when its endpoint is at origin (0,0).
\(\overrightarrow{ \text v} = \sqrt{x^2 + y^2}\)

The starting and ending point of the vector is at certain points (x_{1}, y_{1}) and (x_{2}, y_{2}) respectively.
\(\overrightarrow{ \text v} = \sqrt{(x_2  x_1)^2 + (y_2  y_1)^2}\)
To determine the magnitude of a twodimensional vector from its coordinates,
 Step 1: Identify its components.
 Step 2: Find the sum of the square of each of its components.
 Step 3: Take the square root of the sum so obtained.
Thus, the formula to determine the magnitude of a vector v = (x\(_1\), y\(_1\)) is: \(\overrightarrow{ \text v} = \sqrt{x^2 + y^2}\). This formula is derived from the Pythagorean theorem.
To determine the magnitude of a threedimensional vector from its coordinates,
 Step 1: Identify its components.
 Step 2: Find the sum of the square of each of its components.
 Step 3: Take the square root of the sum so obtained.
Thus, the formula to determine the magnitude of a vector V = (x\(_1\), y\(_1\), z\(_1\)) is: V = \(\sqrt{x_1^2 + y_1^2 + z_1^2}\)
Let us see the applications of the magnitude formula in the following section.
Examples Using Magnitude of a Vector Formula
Example 1: Using magnitude formula, find the magnitude of the vector with \(\overrightarrow{\text u}\) = (2, 5) ?
Solution:
To find: Magnitude of the given vector
Given:
Vector \(\overrightarrow{ \text u}\) = (2,5)
Using magnitude formula,
\(\overrightarrow{ \text u} \)= √(x^{2} + y^{2})
= √(2^{2} + 5^{2})
= √(4 + 25)
\(\overrightarrow{ \text u} \) = 5.385
Answer: Magnitude of the given vector = 5.385
Example 2: Find the magnitude of the vector \(3\hat i + 4\hat j  5\hat k\).
Solution:
To find: Magnitude of the given vector
Given vector \(\bar A = 3\hat i + 4\hat j  5\hat k\)
A = √(3^{2} + 4^{2} + (5)^{2}) =
√(9 + 16 + 25) =√50
=5√2
Answer: Magnitude of the given vector = 5√2
Example 3: Find the magnitude of the vector \(5\hat i  4\hat j + 2\hat k\).
Solution:
To find: Magnitude of the given vector
Given vector \(\bar A = 5\hat i  4\hat j + 2\hat k\)
A =√(5^{2} + (4)^{2} + 2^{2})
√(25 + 16 + 4) = √45
=3√ 5
Answer: Magnitude of the given vector = 3√5
FAQs on Magnitude of a Vector Formula
What Is the Magnitude of a Vector Formula?
The magnitude of a vector formula summarizes the numeric value for a given vector. It is denoted by \(\overrightarrow{ \text v}\). The magnitude of vector formulas are as follows:
 \(A = \sqrt{x_1^2 + y_1^2 + z_1^2}\) for a vector \(\bar A = x \hat i + y \hat j + z \hat k \) its magnitude is:
 \(\overrightarrow{ \text v} = \sqrt{x^2 + y^2}\) when its endpoint is at origin (0,0)
 \(\overrightarrow{ \text v} = \sqrt{(x_2  x_1)^2 + (y_2  y_1)^2}\) when the starting and ending point of the vector is at certain points (x_{1}, y_{1}) and (x_{2}, y_{2})
How To Use the Magnitude of a Vector Formula?
In order to use the magnitude of a vector formula, follow the steps given below
 Step 1: Check for the given parameters.
 Step 2: Put the values in the appropriate formula
For a vector \(\bar A = x \hat i + y \hat j + z \hat k \) its magnitude is \(A = \sqrt{x_1^2 + y_1^2 + z_1^2}\)
The magnitude of a vector when its endpoint is at origin (0,0) then \(\overrightarrow{ \text v} = \sqrt{x^2 + y^2}\)
The starting and ending point of the vector is at certain points (x_{1}, y_{1}) and (x_{2}, y_{2}) then \(\overrightarrow{ \text v} = \sqrt{(x_2  x_1)^2 + (y_2  y_1)^2}\)
What Concept Is Behind the Formula For Calculating the Magnitude of a Vector?
The magnitude of a vector refers to the length or size of the vector. It also determines its direction. The concepts behind these formulas include the Pythagorean theorem and the distance formula, which are used to derive the formula of the magnitude of the vector.
What Is the Magnitude of Vector Formula In Words?
For a given vector with direction ratios along the xaxis, yaxis, and zaxis, the magnitude of the vector is equal to the square root of the sum of the square of its direction ratios.