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Enter coordinates for each component to compute dot, lengths, angle.
2–10
Provide |A|, |B| and included angle to compute dot product.
Vector A
Vector B
Results
If a vector is zero, the angle is undefined.
Step-by-step
Calculated Rows
| # | Timestamp | Mode | Dim | A | B | A·B | |A| | |B| | θ (deg) | θ (rad) |
|---|
Example data table
Click Load to populate inputs.
| Load | Mode | Dim | A | B | |A| | |B| | Angle | Units |
|---|---|---|---|---|---|---|---|---|
| components | 3 | [1, 2, 3] | [4, 5, 6] | — | — | — | — | |
| components | 2 | [3, -4] | [5, 1] | — | — | — | — | |
| magnitudes | — | — | — | 5 | 7 | 33 | deg |
Formula used
Dot product: $\\; \\mathbf{A}\\cdot\\mathbf{B} = \\sum_{i=1}^n a_i b_i$
Lengths: $\\; \\lVert \\mathbf{A} \\rVert = \\sqrt{\\sum a_i^2},\\; \\lVert \\mathbf{B} \\rVert = \\sqrt{\\sum b_i^2}$
Angle: $\\; \\theta = \\arccos\\!\\left(\\dfrac{\\mathbf{A}\\cdot\\mathbf{B}}{\\lVert\\mathbf{A}\\rVert\\,\\lVert\\mathbf{B}\\rVert}\\right)$
Or, with magnitudes and included angle: $\\; \\mathbf{A}\\cdot\\mathbf{B} = \\lVert\\mathbf{A}\\rVert\\,\\lVert\\mathbf{B}\\rVert\\cos\\theta$
How to use
- Choose a mode: Components or Magnitudes + included angle.
- For components: set dimension, enter coordinates for vectors A and B.
- For magnitudes: enter |A|, |B|, and the angle with units.
- Set precision and optionally keep Clamp acos domain enabled.
- Click Compute to see dot product, lengths, and angle.
- Use Download CSV or Download PDF to export the results table.
- Try the Example data table to see typical inputs.
FAQs
It measures how much two vectors align. A positive value indicates acute angle, negative indicates obtuse, and zero implies perpendicular vectors.
The angle formula uses division by both lengths. If either vector is the zero vector, the angle is undefined because division by zero would occur.
Use degrees for everyday angles and radians for calculus or trigonometric identities. The calculator converts and displays both for convenience.
Yes. Set the dimension between 2 and 10. The dot product generalizes to any finite dimension by summing component-wise products.
Floating‑point rounding may push the cosine value slightly beyond ±1. Clamping restricts it to [−1, 1] so arccos remains valid and stable.
Non‑numeric entries, empty fields, or undefined angles can produce NaN. Ensure all inputs are valid numbers and both vectors are non‑zero when computing the angle.
The exports include one row per calculation with timestamp, mode, dimension (if applicable), vectors or magnitudes, dot product, both lengths, and the angle in degrees and radians.