Calculators / Calculus 3 / Unit Vector Calculator

Unit Vector Calculator

Published on: March 1, 2026

This Unit Vector Calculator helps you find the unit vector in the direction of a 3D vector. First find the magnitude of the vector, then divide each component by that magnitude to get a vector of length 1. This follows the unit vector formula used in coordinate geometry and vector algebra. It is a simple way to check answers, understand the method clearly, and practise vector operations step by step.

Step-by-step method

  1. Identify the vector components.
  2. Compute the magnitude |A|.
  3. Use  = A / |A| and simplify.

Formula:

|A|=√a12+a22+a32
A=
A
|A|
=(
a1
|A|
,
a2
|A|
,
a3
|A|
)

Example 1: (3,4,0)

Step 1 - Identify the components.

In this problem: Write the component values.

a1=3, a2=4, a3=0

Step 2a - Write the magnitude formula.

In this problem: Use the magnitude formula.

|A|=√a12+a22+a32

Step 2b - Substitute values.

In this problem: Replace a₁, a₂, a₃ with your values.

|A|=√( 3 )2+( 4 )2+( 0 )2

Step 2c - Solve the squares.

In this problem: Evaluate each squared term.

|A|=√9+16+0

Step 2d - Add the terms.

In this problem: Add inside the square root.

|A|=√25

Step 2e - Simplify |A|.

In this problem: No further simplification is needed.

|A|=5

Step 3a - Write the unit vector formula.

In this problem: Use  = A / |A|.

A=
A
|A|
=(
a1
|A|
,
a2
|A|
,
a3
|A|
)

Step 3b - Substitute values.

In this problem: Substitute the vector components and |A| into the unit vector formula.

A=(
3,4,0
555
)

Step 3c - Simplify the components.

In this problem: Simplify each component (for example, 0/|A| becomes 0).

A=(
3
5
,
4
5
, 0 )

Final answer: Â = (3/5, 4/5, 0)

Example 2: (1/2,0,0)

Step 1 - Identify the components.

In this problem: Write the component values.

a1=
1
2
, a2=0, a3=0

Step 2a - Write the magnitude formula.

In this problem: Use the magnitude formula.

|A|=√a12+a22+a32

Step 2b - Substitute values.

In this problem: Replace a₁, a₂, a₃ with your values.

|A|=√(
1
2
)2+( 0 )2+( 0 )2

Step 2c - Solve the squares.

In this problem: Evaluate each squared term.

|A|=√
1
4
+0+0

Step 2d - Add the terms.

In this problem: Add inside the square root.

|A|=√
1
4

Step 2e - Simplify |A|.

In this problem: No further simplification is needed.

|A|=
1
2

Step 3a - Write the unit vector formula.

In this problem: Use  = A / |A|.

A=
A
|A|
=(
a1
|A|
,
a2
|A|
,
a3
|A|
)

Step 3b - Substitute values.

In this problem: Substitute the vector components and |A| into the unit vector formula.

A=(
1
2
,0,0
1
2
1
2
1
2
)

Step 3c - Simplify the components.

In this problem: Simplify each component (for example, 0/|A| becomes 0).

A=( 1, 0, 0 )

Final answer: Â = (1, 0, 0)