Unit Vector Calculator

Created by Bogna Szyk and Hanna Pamuła, PhD candidate
Reviewed by Steven Wooding
Last updated: Mar 11, 2022

This unit vector calculator will help you transform any vector into a vector of length 1 without changing its direction. If you want to know how to calculate a unit vector's components, look no further! You can obtain the result by dividing the components of any arbitrary vector by its magnitude. Don't worry if you don't know how to find a vector's magnitude, though. This article will give you a step-by-step explanation.

What is a unit vector?

A unit vector is a vector of length equal to 1. When we use a unit vector to describe a spatial direction, we call it a direction vector. In a Cartesian coordinate system, the three unit vectors that form the basis of the 3D space are:

  • (1, 0, 0) — Describes the x-direction;
  • (0, 1, 0) — Describes the y-direction; and
  • (0, 0, 1) — Describes the z-direction.

Every vector in a 3D space is equal to a sum of unit vectors.

Unit vector formula

If you are given an arbitrary vector, it is possible to calculate what the unit vector is along the same direction. To do that, you have to apply the following formula:

  • û = u / |u|

where:

  • û — Unit vector;
  • u — Arbitrary vector in the form (x, y, z); and
  • |u| — Magnitude of the vector u.

You can calculate the magnitude of a vector using our distance calculator or simply by the equation:

  • |u| = √(x² + y² + z²)

Calculating the magnitude of a vector is also a valuable skill for finding the midpoint of a segment.

The unit vector is a useful concept in describing linear transformations. For example, the matrix norm describes how much a unit vector is stretched when multiplied with a matrix.

How to calculate the unit vector

Let's consider an example of a vector u = (8, -3, 5). To calculate the unit vector in the same direction, you have to follow these steps:

  1. Write down the x, y, and z components of the vector. In this case, x₁ = 8, y₁ = -3 and z₁ = 5.

  2. Calculate the magnitude of the vector u:

    |u| = √(x₁² + y₁² + z₁²)

    |u| = √(8² + (-3)² + 5²)

    |u| = √(64 + 9 + 25)

    |u| = √98

    |u| = 9.9

  3. Now that you know the magnitude of the vector u, you probably want to know how to calculate the unit vector. All you have to do is divide each of the initial vector's components by |u|.

    x₂ = x₁ / |u| = 8 / 9.9 = 0.8081

    y₂ = y₁ / |u| = -3 / 9.9 = -0.3031

    z₂ = z₁ / |u| = 5 / 9.9 = 0.5051

  4. Now, write these results in vector form to find the vector û = (0.8081, -0.3031, 0.5051).

  5. You can check whether the result is correct. If it is, the magnitude of your unit vector should be equal to 1.

FAQ

How do I find a unit vector in the same direction?

Divide the original vector by its magnitude. For example, the vector u = (2, 3) has a magnitude of √(2² + 3²) = √13. Therefore the unit vector that has the same direction is û = (2/√13, 3/√13) = (0.5547, 0.832).

Is (1, 1) a unit vector?

No. The length of a unit vector needs to be equal to 1. Calculating the length of (1, 1), we find that √(1² + 1²) = √2 = 1.414, which is not equal to 1.

What is the magnitude of a unit vector?

The magnitude, or length, of a unit vector is 1.

What is unit vector notation?

The notation you use to denote a unit vector to place a circumflex, or "hat" above the lowercase letter representing the vector. For example, for the vector u, its unit vector is written as û.

Bogna Szyk and Hanna Pamuła, PhD candidate
I want to...
normalize vector
Find a unit vector with the same direction as your given vector.
Dimension
2D
unit vector in 2D
Original vector
x
y
Direction angle θ
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Magnitude
💡 The x and y components are squared in the formula for the vector magnitude. Because of this, you can use either negative or positive values and still get the same magnitude.
Unit vector
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y
Magnitude
1
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