Vector Operations

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Vector Operations

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Key Questions

  • Both a magnitude and a direction must be specified for a vector quantity, in contrast to a scalar quantity which can be quantified with just a number. Any number of vector quantities of the same type (i.e., same units) can be combined by basic vector operations.

  • 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.

  • Forces acting on any object change its movement.

    The change depends on the strength and directions of forces and on the way the object moved before forces acted upon it.

    To determine the result of forces acting on an object at any moment we have, first of all, add forces together according to the rules of adding the vectors (the rule of parallelogram). The resulting force is the source of acceleration of an object towards the direction of this force. The absolute value of the acceleration depends on the magnitude of the resulting force and the mass of an object.

    If, for example, the resulting force directed along the way the object is moving, it will continue moving along the same trajectory, but accelerating on its way.

    If the resulting force directed against the movement, the object will decelerate.

    If the resulting force acts at an angle to a trajectory of an object, it will deviate from that trajectory. The degree of deviation depends on the strength and direction of the resulting force and the mass of an object. The greater the mass - the less deviation from the trajectory would be observed.

    The best way to get deeper into this is to solve a few concrete problems with given movement and mass of an object and the forces acting upon it.