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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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