Vector Projection

The Vector Projection calculator computes the resulting vector (W)  that is a projection of vector V onto vector U in three dimensional space. Vector V projected on vector U

INSTRUCTIONS: Enter the following:

• (V) Vector V 
• (U) Vector U

Vector Projection (W): The calculator returns the vector in comma separated form.

The Math / Science

To compute the projection of vector V onto vector U:

1. Compute the magnitude of vector V
2. Compute angle between vectors V and U
3. Compute the unit vector of vector U
4. Compute the scalar (k) associated with the projection which is the magnitude of V times the cosine of the angle between them.
5. Compute the vector resulting in the scalar multiplication of the unit vector of U and the scalar (k).

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