Unit Vector

Unit Vector Calculator

One of the most studied branches of Mathematics is Vector Mathematics in which you have to deal with vectors. In this field, you don’t only deal with numbers but also with directions of the given measurements. Some calculations are common in this field and need to be understood by everyone.

Unit vector calculation is one that has great importance in the real life as well as with educational aspects. It is neither difficult to do so nor easy because you should be aware of vector mathematics. With the help of this unit vector calculation, you can easily make this process easier. This maths calculator can find a unit vector easily using your inserted values for a particular vector.

What is a unit vector?

It is a specific vector that has a magnitude equal to 1 with a specific direction. A unit vector is normally used to show the direction of any measurement or vector. The length of this particular vector must be 1 because of which it is given the name of the unit vector.

It is used widely in Physics and Mathematics to represent the direction of any measurement like Force, Distance, etc. A unit vector is represented by writing a cap (^) on the name of the vector that will be an alphabet.

Unit vector formula

To find a unit vector for any specific vector, we need to use a particular formula. Here is the general formula you should use for finding the unit vector for any particular vector.

a = a/|a|	

Here, the unit vector is on the left side of the equation while the numerator on the right side is the vector and the denominator is the magnitude of the vector that is calculated by using the coordinates of x, y, and z axes.

How to calculate unit vectors with ease?

No doubt, you have learned the formula used to find the unit vector. But you may not know how to find this vector step by step. So, we have shared the steps here by following which you can easily find the unit vector of any given vector.

  • Find the magnitude of the vector by addition of the squares of the coordinates and then taking the square root of the final answer
  • Divide the magnitude by every coordinate of the given vector
  • In the end, you will get the coordinates of the unit vector.

For your better understanding, we have solved an example here. Go through it to learn how this process is completed.

Example 1:

If the vector u has coordinates (5, 6, 9), then what will be the coordinates of the unit vector?


As mentioned earlier, we have to find the magnitude of “u”. For this, we need to use the following formula, So,

|u| = √x2 + y2 + z2	 

From the given data, we know that:

x = 5	
y = 6	
z = 9	


|u| = √52 + 62 + 92	 
= √25 + 36 + 81	 
|u| = 11.92	

Now, we have to use the above formula and divide all coordinates with this magnitude. So, the coordinates for the unit vector will be,

x = 5/11.92	
= 0.42	
y = 6/11.92	
= 0.5	
z = 9/11.92	
= 0.76	   

It means the unit vector for the given vector “u” will be (0.42, 0.5, 0.76).

How to use the unit vector calculator?

As you can see it will be a hectic process to find the unit vector through a manual approach. To make the process simpler and faster, you can use this online tool Calculator’s Bag. Just follow these steps to use this tool.

  • Write the “x”, “y”, and “z” coordinates of the given vector
  • This calculator will automatically show you its magnitude and the coordinates for the unit vector

FAQ | Unit Vector Calculator

What two unit vectors make an angle of 60 with v 8i 6j?

The two vectors that make an angle of 60 with v having coordinates (8, 6) will be (2/5 +√5/10, 3/10 + 2√5/15 ) and (2/5 -√5/10, 3/10 - 2√5/15).

Is (1, 1) a unit vector?

No, (1, 1) is not a unit vector.

Are all unit vectors equal?

No, because unit vectors may have different directions.

Can the unit vector calculator handle negative vectors?

Yes, this calculator can handle either positive or negative vectors.

What is the difference between a position vector and a unit vector?

A position vector shows the position of any point or object and can have any magnitude. But a unit vector can have only 1 magnitude with any direction.