Vector Angle Calculator
Calculate the angle between two vectors using the dot product formula.
Inputs
Angle (degrees)
12.9332
Angle (radians)
0.225726
cos(θ)
0.974632
Step by step
A
= [1, 2, 3]
B
= [4, 5, 6]
A · B
= 32
||A||
= 3.741657
||B||
= 8.774964
cos(θ) = A·B / (||A||·||B||)
= 0.974632
θ
= 12.9332° (0.225726 rad)
How it works
The angle θ between two vectors is found from the dot product formula: cos(θ) = (A·B) / (||A|| · ||B||). The result is always between 0° and 180°. Orthogonal vectors yield 90°, parallel vectors yield 0° or 180°.
Formula
Angle Formula
θ = arccos(A·B / (||A|| · ||B||))
- A
- First vector
- B
- Second vector
- θ
- Angle between them
Frequently Asked Questions
Can the angle be greater than 180°?
No. The angle from the dot product formula is always in [0°, 180°]. For oriented angles in 2D you would need atan2.
What angle do orthogonal vectors make?
Exactly 90°. The cosine of 90° is zero, which means the dot product is zero.