V3 - Angle between vectors

vCalc Reviewed
Equation / Last modified by AndrewBudd on 2018/10/12 16:00
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vCalc.V3 - Angle between vectors

The Angle Between Vectors calculator computes the angle(α) separating two vectors (V and U) V3 - V U.png Vectors U and V in three dimensions in three dimensional space.

INSTRUCTIONS: Enter the following:

  • (V):  Enter the x, y and z components of V separated by commas (e.g. 3,9,1)
  • (U):  Enter the x, y and z components of U separated by commas (e.g. 3,9,1)

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

The Math

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

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