Vector Operations
=================

The vector object has the properties of 3D vectors in math and science.

``r = vector(x,y,z)``  or ``r=vec(x,y,z)`` produce the 3D vector < x, y, z >.  

.. contents::

Vector Addition and Subtraction
-------------------------------

.. code-block::

	a = vector(1,2,3)
	b = vector(10,20,30)
	print(a+b)    # displays < 1, 22, 33 > 

Scalar Multiplication
---------------------

.. code-block::

	a = vector(1,2,3)
	c = 2*a
	print(c)   # displays < 2, 4, 6 >

Magnitude
---------

``mag`` can be invoked as a function or an attribute:

.. code-block::

	a = vector(1,2,3)
	b = vector(10,20,30)
	d = mag(a)  
	print(d)    # displays 3.74166
	e = b.mag
	print(e)    # displays 37.4166

Unit Vector
-----------

There are two names for the function that calculates a unit vector: hat() and norm(). Both can be invoked as functions or attributes:

.. code-block::

	a = vector(1,2,3)
	b = vector(10,20,30)
	f = hat(a)  
	print(f)  # displays < 0.267261, 0.534522, 0.801784 >
	g = norm(a)
	print(g)  # displays < 0.267261, 0.534522, 0.801784 >

	h = a.hat
	i = b.norm

For convenience, ``hat( vec(0,0,0) ) = vec(0, 0, 0)`` to avoid runtime errors.

Dot Product
-----------

Both dot and cross products can be invoked as functions or attributes:

.. code-block::

	a1 = vector(3,1,0)
	b1 = vector(0,2,0)
	j = dot(a,b)
	print(j)   # displays 2
	k = cross(a,b)
	print(k)   # displays < 0, 0, 6 > 

	m = a1.dot(b1)
	n = a1.cross(b1)

Angle Between Vectors
---------------------

The diff_angle() function returns the angle between two vectors in radians:

.. code-block::

	a1 = vector(3,1,0)
	b1 = vector(0,2,0)
	p = diff_angle(a1,b1)
	print(p)  # displays 1.24905

	q = a1.diff_angle(b1)


Projections
-----------

For convenience, the proj() function calculates the vector projection of one vector onto another:

.. code-block::

	A = vector(3,1,0)
	B = vector(0,2,0)
	C = proj(A,B) 
	print(C)   # displays < 0, 1, 0 >

Note that ``proj(A,B) = dot( A, hat(B) ) * hat(B)``

The comp() function calculates the scalar projection of A along B:

.. code-block::

	A = vector(3,1,0)
	B = vector(0,2,0)
	C = comp(A,B) 
	print(C)   # displays 1.

Note that ``comp(A,B) = dot( A, hat(B) )``

Random Vector
-------------

``vector.random()`` produces a vector whose components are random numbers in the range -1 to +1.

Rotating a Vector
-----------------

..  py:function:: myvec2 = rotate( myvec1, angle=a, axis=vector(x,y,z) )
	:noindex:

	:param firstargument: Vector to be rotated (*myvec1* in example above).
	:type firstargument: vector
	:param angle: Angle of rotation in radians.
	:type angle: scalar
	:param axis: Axis of rotation.  Default < 0, 0, 1 >.
	:type axis: vector