Angle Between Vectors Calculator

For example "3, 4" in 2D or "1, 2, 3" in 3D.
Alternative entry: individual components of a.
Alternative entry: individual components of b.
Angle
16.260205 deg
Angle Between Two Vectors
Angle
0.28379411 rad
Angle
18.066894 grad
Angle in the chosen unit
16.260205
Dot product a·b
24.00000000
|a|
5.00000000
|b|
5.00000000
cos θ (cosine similarity)
0.9600000000
|a × b|
7.00000000
Scalar projection of a onto b
4.80000000
Relationship
oblique
Dimensions
2
Worked out from Weisstein, Eric W. "Dot Product." MathWorld — a·b = |a||b| cos θ and 2 other sources.

How this is worked out

7 steps
  1. a·b = (3)(4) + (4)(3) = 24
  2. |a| = √(3² + 4²) = 5
  3. |b| = √(4² + 3²) = 5
  4. cos θ = (a·b) / (|a||b|) = 24 / (5 × 5) = 0.96
  5. θ = arccos(0.96) = 16.26020471° = 0.2837941092 rad
  6. a × b = (0, 0, -7), |a × b| = 7
  7. Scalar projection of a onto b = (a·b)/|b| = 4.8
Cross product a × btable
0, 0, -7
Vector projection of a onto btable
3.84, 2.88
Unit vector âtable
0.6, 0.8
Unit vector b̂table
0.8, 0.6

Results are general information, not financial, tax, legal, medical or engineering advice.