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.
What is Magnitude of a Vector Formula?
Magnitude of a vector \(\bar A \) 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.
For a vector \(\bar A = a \hat i + b \hat j + c \hat k \) its magnitude is:
\[ A = \sqrt{a_1^2 + b_1^2 + c_1^2}\]
Let us solve a few examples to more easily understand the magnitude of a vector formula.
Solved Examples on Magnitude of a Vector Formula

Example 1: Find the magnitude of the vector \(3\hat i + 4\hat j  5\hat k\).
Solution:
Given vector \(\bar A = 3\hat i + 4\hat j  5\hat k\)
\(\begin{align}A &= \sqrt{3^2 + 4^2 + (5)^2} \\&= \sqrt{9 + 16 + 25} \\&= \sqrt{50}\\&=5\sqrt2\end{align}\). 
Example 2: Find the magnitude of the vector \(5\hat i  4\hat j + 2\hat k\).
Solution:
Given vector \(\bar A = 5\hat i  4\hat j + 2\hat k\)
\(\begin{align}A &= \sqrt{5^2 + (4)^2 + 2^2} \\&= \sqrt{25 + 16 + 4} \\&= \sqrt{45}\\&=3\sqrt 5\end{align}\)