V3 - Mixed Product (cross and dot)

The vector Mixed Product (V x U • W) calculator computes the dot product of a vector (W) and the cross product of two other vectors (V and U) in three dimensional space.

INSTRUCTIONS: Enter the following:

• (V): Vector V 
• (U): Vector U 
• (W): Vector W

Mixed Product (m): The calculator returns the mixed product as a real number.

The Math / Science

The Mixed Product of three vectors is the cross product of two vectors creating a third vector that is orthogonal (90 degrees) from both original vectors and then the dot product of the result vector and a third vector.  

The process is as follows:

1. V x U :Cross Product of vectors V and U
2. V x U  • W : Dot product of vectors VxU and W

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

References

• Light and Matter by Benjamin Crowell, Chapter 7.1 Vector Notation