# VectorAngle

VectorAngle[u,v]

gives the angle between the vectors u and v.

# Details • VectorAngle gives an angle in radians.
• For nonzero real vectors the vector angle satisfies .
• For complex vectors the numerator is .

# Examples

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## Basic Examples(2)

The angle between two vectors in 2D:

The angle between two vectors in 3D:

The angle between orthogonal vectors:

## Scope(2)

Use exact arithmetic to compute the vector angle:

Use machine arithmetic:

Use 47-digit precision arithmetic:

Use symbolic vectors:

## Generalizations & Extensions(1)

For complex vectors, the angle returned may be complex:

## Applications(3)

Find when two vectors have the same direction:

Find the area of the triangle, with u and v as two sides:

Plot the area in the triangle formed by the axis and a unit vector in the first quadrant:

Distribution of angles between random vectors with positive entries in 2, 3, 5, and 10 dimensions:

## Properties & Relations(6)

The vector angle satisfies :

The generalization to complex vectors satisfies :

If you rotate a vector u in a plane that includes u, then the vector angle is the rotation angle:

If you rotate it in a plane that does not include u, then the angles differ:

The vector angle is related to the cross product through :

ArcTan of two arguments gives the signed vector angle between the axis and the vector:

Eigenvectors are the vectors for which the angle between and is 0:

## Possible Issues(1)

The angle between the zero vector and any other vector is indeterminate: