Calculators / Calculus 3 / Angle Between Vectors Calculator

Angle Between Vectors Calculator

Published on: March 15, 2026

This Angle Between Vectors Calculator helps you find the angle between two 3D vectors. First find the dot product of the vectors and the magnitude of each vector, then substitute those values into the angle formula. After simplifying, use the inverse cosine to get the final angle. 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 dot product A · B.
  3. Compute the magnitudes |A| and |B|.
  4. Use θ = arccos((A · B) / (|A||B|)) and evaluate.

Formula:

θ=arccos(
A·B
|A||B|
)

Example 1: (1,0,0),(0,1,0)

Step 1 - Identify the components.

In this problem: From the given vectors A = (1, 0, 0) and B = (0, 1, 0), the components are:

a1=1, a2=0, a3=0
b1=0, b2=1, b3=0

Step 2a - Write the dot product formula.

In this problem: Use A · B = a₁b₁ + a₂b₂ + a₃b₃.

A·B=a1b1+a2b2+a3b3

Step 2b - Substitute values.

In this problem: Replace aᵢ and bᵢ with your values.

A·B=( 1 )·( 0 )+( 0 )·( 1 )+( 0 )·( 0 )

Step 2c - Multiply and add.

In this problem: Compute the dot product.

A·B=0+0+0=0

Step 3a - Write the magnitude formula for |A|.

In this problem: Use the magnitude formula for |A|.

|A|=√a12+a22+a32

Step 3b - Substitute values and compute |A|.

In this problem: Substitute the components of A and simplify.

|A|=√( 1 )2+( 0 )2+( 0 )2=1

Step 3c - Write the magnitude formula for |B|.

In this problem: Use the magnitude formula for |B|.

|B|=√b12+b22+b32

Step 3d - Substitute values and compute |B|.

In this problem: Substitute the components of B and simplify.

|B|=√( 0 )2+( 1 )2+( 0 )2=1

Step 3e - Compute |A||B|.

In this problem: Multiply the magnitudes.

|A||B|=1·1=1

Step 4a - Write the angle formula.

In this problem: Use θ = arccos((A · B) / (|A||B|)).

θ=arccos(
A·B
|A||B|
)

Step 4b - Substitute dot product and |A||B|.

In this problem: Substitute A · B and |A||B| into the formula.

θ=arccos(
0
1
)

Step 4c - Evaluate θ.

In this problem: Compute θ in radians.

θ=
1
2
π rad

Step 4d - Approximate θ in radians.

In this problem: Compute a decimal approximation in radians.

θ≈1.57079633 rad

Step 4e - Approximate θ in degrees.

In this problem: Convert the angle to degrees.

θ≈90°

Final answer: θ = pi/2 rad ≈ 1.57079633 rad ≈ 90°

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

Step 1 - Identify the components.

In this problem: From the given vectors A = (1/2, 0, 0) and B = (2, 0, 0), the components are:

a1=
1
2
, a2=0, a3=0
b1=2, b2=0, b3=0

Step 2a - Write the dot product formula.

In this problem: Use A · B = a₁b₁ + a₂b₂ + a₃b₃.

A·B=a1b1+a2b2+a3b3

Step 2b - Substitute values.

In this problem: Replace aᵢ and bᵢ with your values.

A·B=(
1
2
)·( 2 )+( 0 )·( 0 )+( 0 )·( 0 )

Step 2c - Multiply and add.

In this problem: Compute the dot product.

A·B=1+0+0=1

Step 3a - Write the magnitude formula for |A|.

In this problem: Use the magnitude formula for |A|.

|A|=√a12+a22+a32

Step 3b - Substitute values and compute |A|.

In this problem: Substitute the components of A and simplify.

|A|=√(
1
2
)2+( 0 )2+( 0 )2=
1
2

Step 3c - Write the magnitude formula for |B|.

In this problem: Use the magnitude formula for |B|.

|B|=√b12+b22+b32

Step 3d - Substitute values and compute |B|.

In this problem: Substitute the components of B and simplify.

|B|=√( 2 )2+( 0 )2+( 0 )2=2

Step 3e - Compute |A||B|.

In this problem: Multiply the magnitudes.

|A||B|=
1
2
·2=1

Step 4a - Write the angle formula.

In this problem: Use θ = arccos((A · B) / (|A||B|)).

θ=arccos(
A·B
|A||B|
)

Step 4b - Substitute dot product and |A||B|.

In this problem: Substitute A · B and |A||B| into the formula.

θ=arccos(
1
1
)

Step 4c - Evaluate θ.

In this problem: Compute θ in radians.

θ=0 rad

Final answer: θ = 0 rad