Unit Vector Calculator

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.

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.

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.

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 vector's x, y, and z components. 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.