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`V' = rotate("V around U by " alpha )`

Enter a value for all fields

The **Vector Rotation** calculator computes the resulting 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. Enter the x, y and z components of V separated by commas (e.g. 3,9,1) - (
**U**): Rotation axis vector (U). Enter the x, y and z components of U separated by commas (e.g. 3,9,1) - (
**α**): Rotation angle. Enter the angle of rotation. The default is degrees but radians and other units are available.

**Rotated Vector (V'):** The calculator returns the resultant vector (V') in comma separated form.

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.

**k V**- scalar multiplication**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.**(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- Vector Normal to a Plane Defined by Three Points

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

- 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).