# vector

Routines for working with vectors These routines can be used with vectors, as well as with matrices containing a vector in each row.

## Functions

- `vector.angle()` ... Angle between two vectors
- `vector.GramSchmidt()` ... Gram-Schmidt orthogonalization of three points
- `vector.normalize()` ... Normalization of a vector
- `vector.project()` ... Projection of one vector onto another
- `vector.plane_orientation()` ... Orientation of plane defined by three points
- `vector.q_shortest_rotation()` ... Quaternion indicating the shortest rotation from one vector into another.
- `vector.rotate_vector()` ... Rotation of a vector

## Details

Routines for working with vectors These routines can be used with vectors, as well as with matrices containing a vector in each row.

`vector.``GramSchmidt`(p1, p2, p3)

  Gram-Schmidt orthogonalization

  Parameters: | p1 (array (3,) or (M,3)) – coordinates of Point 1 p2 (array (3,) or (M,3)) – coordinates of Point 2 p3 (array (3,) or (M,3)) – coordinates of Point 3
  Returns: | Rmat – flattened rotation matrix
  Return type: | array (9,) or (M,9)

  Example

      >>> P1 = np.array([[0, 0, 0], [1,2,3]])
      >>> P2 = np.array([[1, 0, 0], [4,1,0]])
      >>> P3 = np.array([[1, 1, 0], [9,-1,1]])
      >>> GramSchmidt(P1,P2,P3)
      array([[ 1.        ,  0.        ,  0.        ,  0.        ,  1.        ,
           0.        ,  0.        ,  0.        ,  1.        ],
         [ 0.6882472 , -0.22941573, -0.6882472 ,  0.62872867, -0.28470732,
           0.72363112, -0.36196138, -0.93075784, -0.05170877]])

  Notes

  The flattened rotation matrix corresponds to

  \[\mathbf{R} = [ \vec{e}_1 \, \vec{e}_2 \, \vec{e}_3 ]\]

`vector.``angle`(v1, v2)

  Angle between two vectors

  Parameters: | v1 (array (N,) or (M,N)) – vector 1 v2 (array (N,) or (M,N)) – vector 2
  Returns: | angle – angle between v1 and v2
  Return type: | double or array(M,)

  Example

      >>> v1 = np.array([[1,2,3],
      >>>       [4,5,6]])
      >>> v2 = np.array([[1,0,0],
      >>>       [0,1,0]])
      >>> thLib.vector.angle(v1,v2)
      array([ 1.30024656,  0.96453036])

  Notes

  \[\alpha =arccos(\frac{\vec{v_1} \cdot \vec{v_2}}{| \vec{v_1} | \cdot | \vec{v_2}|})\]

`vector.``normalize`(v)

  Normalization of a given vector

  Parameters: | v (array (N,) or (M,N)) – input vector
  Returns: | v_normalized – normalized input vector
  Return type: | array (N,) or (M,N)

  Example

      >>> thLib.vector.normalize([3, 0, 0])
      array([[ 1.,  0.,  0.]])

      >>> v = [[pi, 2, 3], [2, 0, 0]]
      >>> thLib.vector.normalize(v)
      array([[ 0.6569322 ,  0.41821602,  0.62732404],
         [ 1.        ,  0.        ,  0.        ]])

  Notes

  \[\vec{n} = \frac{\vec{v}}{|\vec{v}|}\]

`vector.``plane_orientation`(p1, p2, p3)

  The vector perpendicular to the plane defined by three points.

  Parameters: | p1 (array (3,) or (M,3)) – coordinates of Point 1 p2 (array (3,) or (M,3)) – coordinates of Point 2 p3 (array (3,) or (M,3)) – coordinates of Point 3
  Returns: | n – vector perpendicular to the plane
  Return type: | array (3,) or (M,3)

  Example

      >>> P1 = np.array([[0, 0, 0], [1,2,3]])
      >>> P2 = np.array([[1, 0, 0], [4,1,0]])
      >>> P3 = np.array([[1, 1, 0], [9,-1,1]])
      >>> plane_orientation(P1,P2,P3)
      array([[ 0.        ,  0.        ,  1.        ],
             [-0.36196138, -0.93075784, -0.05170877]])

  Notes

  \[\vec{n} = \frac{ \vec{a} \times \vec{b}} {| \vec{a} \times \vec{b}|}\]

`vector.``project`(v1, v2)

  Project one vector onto another

  Parameters: | v1 (array (N,) or (M,N)) – projected vector v2 (array (N,) or (M,N)) – target vector
  Returns: | v_projected – projection of v1 onto v2
  Return type: | array (N,) or (M,N)

  Example

      >>> v1 = np.array([[1,2,3],
      >>>       [4,5,6]])
      >>> v2 = np.array([[1,0,0],
      >>>       [0,1,0]])
      >>> thLib.vector.project(v1,v2)
      array([[ 1.,  0.,  0.],
         [ 0.,  5.,  0.]])

  Notes

  \[ \begin{align}\begin{aligned}\vec{n} = \frac{ \vec{a} }{| \vec{a} |}\\\vec{v}_{proj} = \vec{n} (\vec{v} \cdot \vec{n})\end{aligned}\end{align} \]

`vector.``q_shortest_rotation`(v1, v2)

  Quaternion indicating the shortest rotation from one vector into another. You can read “qrotate” as either “quaternion rotate” or as “quick rotate”.

  Parameters: | v1 (ndarray (3,)) – first vector v2 (ndarray (3,)) – second vector
  Returns: | q – quaternion rotating v1 into v2
  Return type: | ndarray (3,)

  Example

      >>> v1 = np.r_[1,0,0]
      >>> v2 = np.r_[1,1,0]
      >>> q = qrotate(v1, v2)
      >>> print(q)
      [ 0.          0.          0.38268343]

`vector.``rotate_vector`(vector, q)

  Rotates a vector, according to the given quaternions. Note that a single vector can be rotated into many orientations; or a row of vectors can all be rotated by a single quaternion.

  Parameters: | vector (array, shape (3,) or (N,3)) – vector(s) to be rotated. q (array_like, shape ([3,4],) or (N,[3,4])) – quaternions or quaternion vectors.
  Returns: | rotated – rotated vector(s)
  Return type: | array, shape (3,) or (N,3)

  Notes

  \[q \circ \left( {\vec x \cdot \vec I} \right) \circ {q^{ - 1}} = \left( {{\bf{R}} \cdot \vec x} \right) \cdot \vec I\]

  More info under http://en.wikipedia.org/wiki/Quaternion

  Examples

      >>> mymat = eye(3)
      >>> myVector = r_[1,0,0]
      >>> quats = array([[0,0, sin(0.1)],[0, sin(0.2), 0]])
      >>> quat.rotate_vector(myVector, quats)
      array([[ 0.98006658,  0.19866933,  0.        ],
             [ 0.92106099,  0.        , -0.38941834]])

      >>> quat.rotate_vector(mymat, [0, 0, sin(0.1)])
      array([[ 0.98006658,  0.19866933,  0.        ],
             [-0.19866933,  0.98006658,  0.        ],
             [ 0.        ,  0.        ,  1.        ]])