This Unit Vector Calculator will allow you to convert any vector into a single-length vector without affecting its direction. Look no further if you want to learn how to compute the components of a unit vector. Furthermore, by dividing the components of any given vector by its magnitude, you may get the result. So don’t worry if you don’t know how to calculate the magnitude of a vector.

What is a unit vector? Unit Vector Definition

Vectors are geometrical objects with magnitude and direction. A vector has a beginning point and a terminal point, which reflects the point’s end location. Addition, subtraction, and multiplication are some of the mathematical operations that we may use on vectors. We define a unit vector as a vector with a magnitude of one. When we divide any vector by the magnitude of another vector, we get a unit vector. A direction vector also refers to a unit vector.

Unit vectors, represented by a, are vectors with a magnitude equal to one. Unit vectors have a length of one. We see the unit vectors to indicate a vector’s direction. The direction of a unit vector is the same as the supplied vector, but the magnitude is one unit.

Unit vector formula

Indication of vectors is with an arrow a, which signifies a unit vector because they have both magnitude (Value) and direction. We divide the magnitude of a vector by its unit vector to determine its unit vector. Any vector usually represents the coordinates x, y, and z.

If you have an arbitrary vector, you can figure out the unit vector in the same direction. To do this, use the following formula: where:

û = u / |u|

  • |u| is the magnitude of the vector u,
  • u is an arbitrary vector in the form (x, y, z), and
  • û is the unit vector.

How to calculate Unit Vector?

Consider the following example of a vector:

u = (8, -3, 5).

We need to follow these procedures to determine the unit vector in the same direction.

  1. Make a list of the vector’s x, y, and z components.
    x1 = 8,
    y1 = -3, and
    z1 = 5 in this situation.
  2. Calculate the vector u’s magnitude.

|u| = √(x₁² + y₁² + z₁²)

= √(8² + (-3)² + 5²) = √(64 + 9 + 25)

= √98 = 9.9

  • You presumably want to know how to compute the unit vector now that you know the magnitude of the vector u. All you have to do is divide each component of the starting vector 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

  • Also, to find the vector û =, express these findings in a vector form (0.8081, -0.3031, 0.5051). You can verify that the outcome is correct. If that’s the case, the magnitude of your unit vector should be 1.

Example – how to find unit tangent vector?

Let v(t) = r'(t) be the velocity vector and r(t) be a differentiable vectorvalued function. We define the unit tangent vector as the unit vector in the velocity vector’s direction.

T(t)  = v(t) / ||v(t)||

r(t)  =  t i + et j – 3t2 k     

v(t)  =  r'(t)  =  i + et j – 6t k     

unit vector - tangent vector


How to find a unit vector in the same direction?

Simply divide your vector by its magnitude to obtain a unit vector with the same direction. Consider a vector with a magnitude of |v|, such as v = (1,4). We may acquire the unit vector v by dividing each component of vector v by |v|, which is in the same direction as v.

Are unit vectors always positive?

In two dimensions, the standard unit vectors. The vectors are unaffected by mouse movement since they always point in the positive direction of their respective axis.

Are unit vectors always perpendicular?

The origin is centered on the unit vectors. They are ALWAYS perpendicular to each other.

Can unit vectors be fractions?

The unit vector may now be found using the magnitude: The unit vector in bracket format is: Because the question did not call for decimal values, it is advisable to leave the numbers in the vector as fractions.

Can a unit vector be more than 1?

Vectors with a magnitude of exactly one unit are known as unit vectors. They are quite beneficial for a variety of reasons. The unit vectors [0,1] and [1,0] are sometimes in combinations to generate any other vector.

Do unit vectors have a direction?

Vector quantities have a magnitude and a direction. However, there are situations when simply the direction of the vector is of relevance, rather than the magnitude. In such instances, vectors are frequently “normalized” to be of unit length for simplicity.

What is the unit vector in physics?

Vectors with a magnitude of exactly one unit are known as unit vectors. They are quite beneficial for a variety of reasons. Sal Khan was the one who came up with the idea.

What is the cross product of two unit vectors?

The area of the parallelogram formed by two vectors, we often call the cross product in geometry. We use the term “cross product” originates from the symbol to symbolize this operation: a huge diagonal cross (x). This product is also known as the vector product since it contains magnitude and direction.