3D Vector Angle Calculator
3D Vector Angle Calculator is a mathematics calculator that helps you calculate 3d vector angle values from your input data. The formula used is: θ = arccos(v₁·v₂ / (|v₁||v₂|)); |v×w| = √((y1z2−z1y2)²+(z1x2−x1z2)²+(x1y2−y1x2)²). Ensure all inputs share the same unit system (e.g. all metres or all feet) unless the formula converts between units.
Formula
θ = arccos(v₁·v₂ / (|v₁||v₂|)); |v×w| = √((y1z2−z1y2)²+(z1x2−x1z2)²+(x1y2−y1x2)²)
Example Calculation
Inputs
- Vector 1 — X: 1
- Vector 1 — Y: 2
- Vector 1 — Z: 3
- Vector 2 — X: 4
- Vector 2 — Y: 5
- Vector 2 — Z: 6
For vectors (1, 2, 3) and (4, 5, 6), the dot product is 4 + 10 + 18 = 32, with magnitudes √14 ≈ 3.742 and √77 ≈ 8.775. The angle is cos⁻¹(32 ÷ 32.83) ≈ 12.93°, and the cross product magnitude is about 7.35.
Frequently asked questions
How do you find the angle between two 3D vectors?
Divide the dot product by the product of the magnitudes and take the inverse cosine: θ = arccos[(v₁·v₂) ÷ (|v₁||v₂|)].
What does the cross product magnitude tell me?
|v₁×v₂| equals |v₁||v₂|·sinθ and represents the area of the parallelogram the two vectors span. It is zero when the vectors are parallel.
What units should I enter?
Enter each vector's X, Y and Z components in any consistent unit. The angle is returned in degrees.
What if a vector is zero?
A zero vector has no direction, so the angle is undefined; the calculator flags this case.
How accurate are the results?
The dot product, magnitudes, and angle are computed exactly and rounded only for display.
How can I verify the calculation manually?
Use the displayed formula and work through the numbers step by step. If your manual result differs slightly, check rounding and unit conversions first.
Can I use this for planning and budgeting?
Yes. Run best-case, expected, and worst-case inputs to compare outcomes and make safer planning decisions.