# Mathematical concepts

## Scalars

Scalar is simply a mathematical term for a value that is not
a compound value. e.g. *3*. Often mathematicians will talk of
multiplying a vector by a scalar. This simply means that each
component of the vector is multipled by the scalar.

## Vector multiplication

Vectors can be multiplied in two different ways. Refer the the
following figure in the following two sections.

## Dot Product

The dot product of a pair of vectors **u** = (u_{x},
u_{y},u_{z}) and **v** = (v_{x},
v_{y},v_{z}) is defined to be:

**u**·**v** = u_{x}v_{x} +
u_{y}v_{y} + u_{z}v_{z}

This is not a vector, it is a scalar. It is also the case that:

**u**·**v** = |**u**||**v**|cos
&theta

where *&theta* is the angle between the vectors and
*|***w**| is the magnitude of a vector **w**.

In the case of unit vectors which have a magnitude of one one can
see clearly that **u**·**v** = cos
&theta.

It is thus, quite easy to find the angle between two vectors using
the dot product and you will see that we have done so in module
`Physics`

.

## Cross product

Unlike the dot product, the cross product of two vectors is
another vector. This vector points in a direction that is
perpendicular to the other two. Obviously there are two ways for this
to occur. If we view the two vectors as being in the same plane the
resulting vector could point out of either the front or the back
face.

Which direction it does point depends on the relative directions of
the two vectors. If the second vector is clockwise with respect
to the first in the plane then the vector points towards us. Otherwise
it points away from us. This is illustrated in the following figure
(where **u** x **v** = **w**):

In the left diagram, **v** is clockwise with
respect to **u** and so **w** points towards us. On the right,
**v** is anti-clockwise with respect to **u** and so **w**
points away from us.

Now for the definition of the cross product:

**u** x* ***v** = |**u**||**v**|sin &theta
**n**

where *&theta* is the angle between **u** and
**v** and *n* is a unit vector in the direction
perpendicular to both **u** and **v** subject
to the figure above.

It is also the case that:

**u** x* ***v** = (u_{y}v_{z} - u_{z}v_{y}, u_{z}v_{x} - u_{x}v_{z}, u_{x}v_{y} - u_{y}v_{x})

## Planes and rotations

Another way to think of **n** is as the vector that
*defines* the plane in which **u** and
**v** lie. This is particularly useful if you wish to
rotate a vector within a plane. We have provided a function called
`rotateInPlane`

in module (you guessed it) `Physics`

which takes a unit vector
defining a plane, a vector to be rotated, and an angle and rotates the
vector in that plane. See the module for more details.