Enter the X, Y, and Z coordinates of your vector to calculate the equivalent unit vector as a ratio of the magnitude of that vector. This calculator also calculates the magnitude of the original vector and the angle of the vector.

Unit Vector Formula

The formula for the unit vector is as follows:

u = U / |U|
  • Where u is the unit vector,
  • U is the original vector,
  • and |U| is the magnitude of the original vector

To calculate the unit vector, divide the original vector by the magnitude of the original vector.

Unit Vector Definition

A unit vector is the equivalent vector of an original vector that has a magnitude of 1. In other words, it has the same direction as your original vector but the total magnitude is equal to one. Since the unit vector is the original vector divided by magnitude, this means that it can be described as the directional vector. Therefore, if you have the direction vector and the magnitude, you can calculate the actual vector.

How to calculate a unit vector

We will now take a look at an example of how you can calculate a unit vector from a normal vector. Let’s take a vector u = (5,-4,2).

  1. First, you must calculate the magnitude of the vector. This is done through the following formula. |u| = √(x₁² + y₁² + z₁²)
  2. Plugin the values into the formula above, and you should get 6.708.
  3. Next, you need to divide each unit vector point by the magnitude. This is because the unit vector has a magnitude of 1. Dividing the original vector by its magnitude achieves that rule.
  4. This should yield X = .706, Y= – .596, Z = .298
  5. Check the result with the calculator above.

Now you know how to calculate a unit vector, but what happens if the vector is given as two points and neither is positioned at the origin? If this is the case, the steps for solving the unit vector are a little different. Let’s take a vector with points (7,8,9) and (2,5,1)

  1. First, you must understand that a vector, as mentioned earlier, describes a direction and magnitude. Since both these values are independent of position, you can manipulate a vector to re-position it at the origin
  2. Now that you know this, you can place the vector at the origin simply by subtracting one vector from the other. In this case (7,8,9) – (2,5,1) = (5,3,8). You just subtract each coordinate point from the other.
  3. Now that you have the vector placed at the origin, complete steps 1-5 from above and you have your unit vector!

Are unit vectors dimensionless?

Unit vectors are dimensionless and unitless. They are a dimensionless representation of the direction of a vector with a magnitude of 1. Multiplying the unit vector by the magnitude will yield the original vector.

Are unit vectors always positive?

A unit vector can have negative components if there are negative components in the original vector. For example, if there is a vector of units (-2,-2) then the unit vector would be (-.707,-.707).

Are unit vectors always perpendicular?

Unit vectors are not always perpendicular, in fact, unit vectors are always parallel and tangent to the original vector.

Can unit vectors be fractions?

Unit vectors can be expressed as fractions. In the example above of a unit vector of (-.707,-.707), these could be expressed as the fractions (-707/1000,-707/1000).

Can a unit vector be more than 1?

No, a unit vector must have a magnitude of 1 to be considered a unit vector. Also, no individual component of the unit vector can be more than or equal to 1.

Do unit vectors have direction?

Yes, unit vectors have a direction equal to the original vector, but with a magnitude of one. That is they have the same angle as the original vector.

Do unit vectors have magnitude?

Yes, unit vectors do have a magnitude. Its magnitude is always equal to 1.

What is the unit vector along i j?

A unit vector along I j is a unit vector whose original vector is described in the I and j space. Often time the i and j space can be substituted for x and y.


What is a unit vector?

A unit vector is the equivalent vector of an original vector that has a magnitude of 1