Angle between Vectors

The Angle Between Vectors calculator computes the angle(α) separating two vectors (V and U)  in three dimensional space.

INSTRUCTIONS: Enter the following:

• (V):  Vector V
• (U):  Vector U

Angle Between Vectors (α): The calculator returns the angle (α) between the two vectors in degrees and radians.  However, these can be automatically converted into compatible units via the pull-down menu. 

The Math / Science

The angle between vectors formula lets the user enter two three-dimensional vectors (V and U)  with X, Y and Z components (Euclidean 3-space vectors).   

     α = acos(`hatU * hatV`)

where:

• α = angle between `vecV` and `vecU`
• `hatU` = unit vector for `vecU`
• `hatV` = unit vector for `vecV`

To calculate the angle between two vectors:

1. calculate the unit vectors associated with vector V and vector U.  To do that,
   1. compute the magnitude of the vectors  and then
   2. do a scalar multiplication for each of the vectors where the scalar(k) is the inverse of the vector's magnitude.
2. calculate the dot product of the unit vectors
3. calculate the arc-cosine of that dot product to calculate the angle between the vectors in radians.
4. converts radians to degrees. 

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