# Unit Vector Formula

A unit vector formula is used to find the unit vector of the given vector. The given vector is divided by the magnitude of the vector, to obtain the unit vector. The unit vector has all the same vector components of the given vector but has a magnitude of one. The unit vector formula uses the concept of the magnitude of the vector.

## What is Unit Vector Formula?

The unit vector \(\hat A \) is obtained by dividing the vector \(\vec A \) with its magnitude |\( \vec{A}\)|. The unit vector has the same direction coordinates as that of the given vector.

### \[ \hat A = \dfrac{\vec A}{|\vec{A}|}\]

**Break down tough concepts through simple visuals.**

Let us try out a few examples to understand how to use unit vector formula.

## Solved Examples on Unit Vector Formula

**Example 1: Find the unit vector of \(3\hat i + 4\hat j - 5\hat k\).**

**Solution:**

Given vector \(\vec A = 3\hat i + 4\hat j - 5\hat k\)

\(\begin{align}|\vec{A}| &= \sqrt{3^2 + 4^2 + (-5)^2} \\&= \sqrt{9 + 16 + 25} \\&= \sqrt{50}\\&=5\sqrt2\end{align}\)

\(\begin{align}\hat A &= \frac{1}{|\vec{A}|}.\vec A \\&= \frac{1}{5\sqrt2}.(3 \hat i + 4\hat j - 5\hat k)\end{align}\)

**Answer:** Hence the unit vector is \( \frac{1}{5\sqrt2}.(3 \hat i + 4\hat j - 5\hat k) \).

**Example 2: Find the vector of magnitude 8 units and in the direction of the vector \( \hat i - 7\hat j + 2\hat k\).**

**Solution:**

Given vector \(\vec A = \hat i - 7\hat j + 2\hat k \).

\(\begin{align}|\vec{A}| &= \sqrt{1^2 + (-7)^2 + 2^2} \\&= \sqrt{1 + 49 + 4} \\&= \sqrt{54}\\&=3\sqrt6\end{align}\)

\(\begin{align}\hat A &= \frac{1}{|\vec{A}|}.\vec A \\&= \frac{1}{3\sqrt6}.(\hat i - 7\hat j + 2\hat k)\end{align}\)

**Answer:** Therefore the vector of magnitude 8 units = \(\frac{4\sqrt6}{9}.(\hat i - 7\hat j + 2\hat k)\)