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Angle between Vectors

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α = acos (\hat{V} \cdot \hat{U})

$alpha = "acos"(hatV*hatU)$

(V) Vector V

$(V)"Vector V"$

(U) Vector U

$(U)"Vector U"$

Tags

The Angle Between Vectors calculator computes the angle(α) separating two vectors (V and U) /attachments/969993e0-96b7-11e4-a9fb-bc764e2038f2/VectorAngle.JPG 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( $\hat{U} \cdot \hat{V}$ $hatU * hatV$ )

where:

α = angle between $\vec{V}$ $vecV$ and $\vec{U}$ $vecU$
$\hat{U}$ $hatU$ = unit vector for $\vec{U}$ $vecU$
$\hat{V}$ $hatV$ = unit vector for $\vec{V}$ $vecV$

To calculate the angle between two vectors:

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.
calculate the dot product of the unit vectors
calculate the arc-cosine of that dot product to calculate the angle between the vectors in radians.
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