Quaternion Vector Rotation Calculator
Rotate a numeric 3D vector using a normalized wxyz quaternion or axis-angle input and inspect its equivalent matrix.
YOUR CALCULATION
Result
—
Calculation steps
Work through one step at a time or reveal the complete calculation. The steps use your current inputs.
How to calculate
Normalize the quaternion, apply q(0,v)conjugate(q) and report the active rotation matrix acting on a column vector.
v′=v+2w(u×v)+2u×(u×v), for unit q=(w,u)
Worked example
Axis (0,0,1), angle +90° and vector (1,0,0) rotate to (0,1,0), up to floating-point rounding. Replacing q with −q gives the same rotation.
Assumptions and input details
All quaternion/axis/vector entries are numeric and nonzero quaternion/axis norms are required. Component order is explicitly w,x,y,z. Axis-angle is in degrees and rotations are active, right handed, acting on column vectors. No Euler order, translation, projection or handedness conversion is inferred. Scaling a nonzero quaternion is removed by normalization.
Using this calculator
Can I enter my own problem?
Yes. Replace the example values with your own inputs. The result, calculation steps, tables and applicable diagrams update from those inputs. Blank or invalid values clear the result; example values are never silently substituted for missing inputs.
Are the results exact?
Results use floating-point arithmetic and displayed rounding. Read the formula, assumptions and any stated data precision before interpreting the result.
Is my input uploaded?
The calculation runs in your browser. No account, file upload or external AI service is required. Inputs are not saved between page reloads.
Reflect, rotate or translate a point in the coordinate plane. Keep exact fractional coordinates and compare the original point with its image on equal-scale axes.
Enter two three-component vectors to calculate their sum, difference, dot product, cross product, magnitudes and angle, with an XY-projection diagram.
Fit a planar homography to four source/destination point pairs and map a query coordinate with reconstruction residuals.
Reserved advertising space.