Vector operations include
1.) Vector Addition
#\vec A + \vec B = \langle A_x + B_x, A_y+B_y,A_z+B_z\rangle#
The components of the sum are the sums of the respective components of the addends.
2.) The dot product, also called the scalar product:
#\vec A\cdot \vec B = A_xB_x+A_yB_y+A_zB_z#
The magnitude of the dot product is also given by
#|\vec A\cdot \vec B|=|\vec A||\vec B|\cos\theta#, where #\theta# is the angle between the two vectors.
3.) The cross product or vector product:
#\vec A\times \vec B = \langle A_yB_z-A_zB_y, A_zB_x-A_xB_z,A_xB_y-A_yB_x\rangle#
The magnitude of the cross product is also given by
#|\vec A\times \vec B| = |\vec A||\vec B|\sin\theta#, where #\theta# is the angle between the two vectors.
There are also three basic vector derivative operators. They are gradient, divergence and curl.
The gradient operator is an operator which returns a vector pointing in the direction of steepest increase of a scalar function.
#\vec \nabla f = \langle \partial_x f, \partial_y f, \partial_z f\rangle#.
The divergence operator is a derivative operator which acts on a vector valued function (a vector field), and relates to the change of that vector field in the direction that it points. Its result is a scalar:
#\vec \nabla \cdot \vec A = \partial_x A_x + \partial_y A_y+ \partial_z A_z#
Divergence is positive in regions where the field diverges or spreads out. It is negative in regions where the field converges. A field with no divergence anywhere (such as the magnetic field) is said to be divergenceless.
Curl is the vector derivative that quantifies change in a vector field in a direction perpendicular to the direction the vector field itself is pointing.
#\vec \nabla \times \vec A = \langle \partial_yA_z-\partial_zA_y, \partial_zA_x-\partial_xA_z,\partial_xA_y-\partial_yA_x\rangle#.
There is a mnemonic for remembering this formula that uses the determinant of a matrix, but the markup on this site will not allow me to write it here.
You can think of curl like this. Put a little pinwheel in a vector field. Imagine that field as the flow of water. The curl is a vector that points along the direction of the axis of the pinwheel in the direction it would spin at each point in the vector field, in a right handed sense.