`vecV' = rotate(vecV " around " vecU " by " alpha)`

Enter a value for all fields

The Vector Rotation calculator computes the resulting 3D vector created by rotating a base vector (V) about a rotation vector (U) by an angle(α).

INSTRUCTIONS: Enter the following:

• (V):  Base vector (V) to be rotated
• (U):  Rotation axis vector (U) 
• (α):  Rotation angle
• (n):  Number of Significant Digits

Rotated Vector (V'): The calculator returns the resultant vector (V') 

The Math / Science

The Vector Rotation formula uses quaternions to compute the resulting vector from the specified rotation.  It uses the rotation of axis (U) and the rotation angle (α) to compute the quaternion of rotation (q).  It then uses the quaternion vector rotation formula as follows:

           V' = q⋅V⋅q^*
where:

• V' is the rotated resultant vector.
• q is the quaternion of rotation
• q^* is conjugate of the quaternion of rotation.

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3D Vector Functions

• k⋅V - scalar multiplication
• V/k - scalar division
• V / |V| - Computes the Unit Vector
• |V| - Computes the magnitude of a vector
• U + V - Vector addition
• U - V - Vector subtraction
• |U - V| - Distance between vector endpoints.
• |U + V| - Magnitude of vector sum.
• V • U - Computes the dot product of two vectors
• V x U - Computes the cross product of two vectors
• V x U • W - Computes the mixed product of three vectors
• Vector Angle - Computes the angle between two vectors
• Vector Area - Computes the area between two vectors
• Vector Projection - Compute the vector projection of V onto U.
• Vector Rotation - Compute the result vector after rotating around an axis.
• Vector Components 3D - Returns a vector's magnitude, unit vector, spherical coordinates, cylindrical coordinates and angle from each axis.
• (ρ, θ, φ) to (x,y,z) - Spherical to Cartesian coordinates
• (x,y,z) to (ρ, θ, φ) - Cartesian to Spherical coordinates
• (r, θ, z) to (x,y,z) - Cylindrical to Cartesian coordinates
• (x,y,z) to (r, θ, z) - Cartesian to Cylindrical coordinates
• (x,y) to (r, θ) - Cartesian to Polar
• (r, θ) to (x,y) - Polar to Cartesian 
• Vector Normal to a Plane Defined by Three Points

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Quaternion Calculators

• Quaternion Addition
• Quaternion Subtraction
• Quaternion Multiplication
• Quaternion Magnitude
• Quaternion Versor
• Quaternion Conjugate
• Quaternion Inverse
• Quaternion of Rotation
• Vector Rotation

References

• Light and Matter by Benjamin Crowell, Chapter 7.1 Vector Notation
• An Engineer's Approach to Quaternions* (Damon D. Ostrander October 1998)  See Quaternions RevA.pdf (attached).