Unit Vector Calculator

Method

Dimension of a vector

Unit Vector

An online unit vector calculator helps you determine the components of any vector of length 1 without changing its direction. You can also calculate the angle and magnitude of a vector using this calculator. Below, we explain how to find unit vectors, their magnitudes, and provide simple definitions.

What is a Unit Vector?

Vectors with a magnitude equal to 1 are called unit vectors, denoted by a “^” symbol, e.g., ^A. They are sometimes referred to as direction vectors or the multiplicative identity of a vector. Unit vectors indicate the direction of a vector and always have a length of 1.

Unit Vector Formula

A unit vector is obtained by dividing a vector by its magnitude. Unit vectors can be represented in 2D or 3D coordinates as follows:

Bracket notation: u = (x, y, z)
Component notation: u = x î + y ĵ + z k̂

The magnitude of a vector is:

|u| = √(x² + y² + z²)

Thus, the unit vector is:

û = u / |u|

Bracket Format:

û = (x, y, z) / √(x² + y² + z²) = (x/√(x²+y²+z²), y/√(x²+y²+z²), z/√(x²+y²+z²))

Component Format:

û = (x î + y ĵ + z k̂) / √(x² + y² + z²) = (x/√(x²+y²+z²)) î + (y/√(x²+y²+z²)) ĵ + (z/√(x²+y²+z²)) k̂

An online Unit Tangent Vector Calculator can help find the tangent vector of a vector-valued function at given points.

Derivation of the Unit Vector Formula

û = Unit vector in the direction of v
v = Vector with any magnitude and direction
|v| = Magnitude of vector v
x, y, z = Components along x, y, z axes
î, ĵ, k̂ = Unit vectors along x, y, z axes

Formula in component form:

û = v / |v| = (x î + y ĵ + z k̂) / √(x² + y² + z²) = (x/|v|) î + (y/|v|) ĵ + (z/|v|) k̂

How to Find a Unit Vector

You can calculate a unit vector manually or using an online calculator. Steps for manual calculation:

Example:

Find the unit vector of u = (8, -3, 5).

Solution:

Step 1: Identify vector components: a1 = 8, b1 = -3, c1 = 5

Step 2: Magnitude of u:

|u| = √(8² + (-3)² + 5²) = √(64+9+25) = √98 ≈ 9.9

Step 3: Divide each component by magnitude:

a2 = 8 / 9.9 ≈ 0.8081
b2 = -3 / 9.9 ≈ -0.3031
c2 = 5 / 9.9 ≈ 0.5051

Step 4: Unit vector:

û = (0.8081, -0.3031, 0.5051)

Example 1: 2D Unit Vector

Vector: v = (3, 4)

Magnitude: |v| = √(3² + 4²) = 5

Direction angle: θ = arctan(4/3) ≈ 53.13°

Unit vector: ê = (3/5, 4/5) = (0.6, 0.8)

Example 2: Missing 3D Component

Given x = 0.9, y = 0.4. Find z for unit vector:

x² + y² + z² = 1 → 0.9² + 0.4² + z² = 1 → z² = 0.03 → z ≈ 0.173

Unit vector: û = (0.9, 0.4, 0.173)

Spherical Coordinate Unit Vectors

r̂ = sinθ cosφ î + sinθ sinφ ĵ + cosθ k̂

θ̂ = cosθ cosφ î + cosθ sinφ ĵ - sinθ k̂

φ̂ = -sinφ î + cosφ ĵ

How the Unit Vector Calculator Works

Input:

Select vector type and dimensions (2D or 3D)
Enter vector components
Click "Calculate Unit Vector"

Output:

Magnitude of vector
Direction angle
Unit vector

FAQ

Does a unit vector have units?

No. Unit vectors are dimensionless and indicate only direction.

What is a polar vector?

A polar vector has a specific point of origin. Velocity is an example.

Magnitude of the zero vector?

The zero vector has magnitude 0, e.g., stationary object's velocity.

When are two vectors equal?

Two vectors are equal if they have the same magnitude and direction.

Are unit vectors always perpendicular?

No. They follow the direction of the original vector; they may be parallel or tangent.

Conclusion

The unit vector calculator is useful for computing the unit vector, magnitude, and direction of any vector. It is valuable for students and professionals in physics, mathematics, and engineering.

References

Sources: Wikipedia – Unit vector, Cartesian coordinates; Toppr – Unit vector formula; Vedantu – Unit tangent and normal vectors, spherical coordinate system.