Math

Angle Between Two Vectors Calculator

Find the angle between two 3D vectors.

a x

a y

a z

b x

b y

b z

Calculated Output

Angle (°) 45

Angle (rad) 0.7854

Formula: θ = arccos((a · b) ÷ (|a| |b|))

Step-by-Step Breakdown

1. Values used:

2. a x = 1

3. a y = 0

4. a z = 0

5. b x = 1

6. b y = 1

7. b z = 0

9. Angle = 45°

10. Angle = 0.7854rad

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The math behind it

cosθ = (a · b) ÷ (|a| |b|), so θ = arccos of that. A result of 90° means the vectors are perpendicular.

(1,0,0) and (1,1,0) → 45°.

What angle means perpendicular?

90°, which happens exactly when the dot product is zero.