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Unit Vector Calculator
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A unit vector $\hat{U}$ is a vector whose magnitude is 1 and is represented as:
$\hat{U}$ = $\frac{U}{|U|}$
where U is a given vector and |U| is the magnitude or norm of the given vector.
Unit Vector Calculator calculates the magnitude of the given vector and finds whether the given vector is Unit vector or not.

## Steps for Unit Vector Calculator

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Step 1 :

Read the given vector U which will be of the form xi + yj + zk, where x, y and z are the components of the given vector.

Step 2 :

To find the magnitude of the vector use the formula
|U| = $\sqrt{x^{2} + y^{2} + z^{2}}$

Step 3 :

The Unit vector is given as: $\hat{U}$ = $\frac{U}{|U|}$
if the value of vector is 1, then it is a unit vector or else it is not.

## Problems on Unit Vector Calculator

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1. ### Find the unit vector U in the direction of vector U = 6i - 2j + 5k.

Step 1 :

Given the vector is U = 6i - 2j + 5k,

where x = 6

y = -2

z = 5.

Step 2 :

Magnitude of the given vector is |U| = $\sqrt{x^{2} + y^{2} + z^{2}}$

= $\sqrt{6^{2} + 2^{2} + 5^{2}}$

= $\sqrt{64}$

= 8.

Step 3 :

The Unit vector is given as

U = $\frac{U}{|U|}$

U = $\frac{6i - 2j + 5k}{\sqrt{64}}$
$\neq$ 1.

Answer  :

Hence the given vector is not unit vector.

2. ### Find the unit vector U in the direction of vector U = 2i - 2j + k.

Step 1 :

Given the vector is U = 2 $\hat{i}$ - 2 $\hat{j}$ + $\hat{k}$,

where x = 2

y = -2

z = 1.

Step 2 :

Magnitude of the given vector is |U| = $\sqrt{x^{2} + y^{2} + z^{2}}$

= $\sqrt{2^{2} + 2^{2} + 1^{2}}$

= $\sqrt{9}$

= 3.

Step 3 :

The Unit vector is given as

U = $\frac{U}{|U|}$

U = $\frac{2 \hat{i} - 2 \hat{j} + \hat{k}}{\sqrt{9}}$
$\neq$ 1.

Answer  :

Hence the vector 2 $\hat{i}$ - 2 $\hat{j}$ + $\hat{k}$ is not a unit vector.

3. ### Find the unit vector U in the direction of vector U = 0 $\hat{i}$ - 0 $\hat{j}$ + $\hat{k}$.

Step 1 :

Given the vector is V = 0 $\hat{i}$ - 0 $\hat{j}$ + $\hat{k}$,

where x = 0

y = 0

z = 1.

Step 2 :

Magnitude of the given vector is |U| = $\sqrt{x^{2} + y^{2} + z^{2}}$

= $\sqrt{0^{2} + 0^{2} + 1^{2}}$

= $\sqrt{1}$

= 1.

Step 3 :

The Unit vector is given as

$\vec{U}$ = $\frac{U}{|U|}$

$\vec{U}$ = $\frac{0i - 0j + k}{1}$
= 1.

Answer  :

Hence the given vector U = 0 $\hat{i}$ - 0 $\hat{j}$ + $\hat{k}$ is a unit vector.

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