Due Thursday 27th January

Formats and List operations (Vectors)

**Formating**

From the toolbar do the following: ``format'' followed by ``style'',
the dropdown menu offers you lots of
options. ``input'' is the default, and is the ``format''
in which you do calculations. However *Mathematica* also allows
various fonts and styles of text input. In this problem set
you should include a text ``cell'' before each problem which
identifies the problem (e.g. Problem 1), and a text cell at the
top of the page with your name and the number of the worksheet
(e.g. ``worksheet 2'').

Last week we did derivatives and integrals using the
full *Mathematica* commands. However many of these commands
may be entered from ``palettes''. To activate a palette,
from the toolbar do the following: ``file'' followed by
``palettes''. You have several options: ``basic input'' is
one I like.

The reference material for this worksheet is sections
1.2.1 - 1.2.6 of
the practical introduction to mathematica. Your should read this material
as you work through the excersizes below.

**Lists and Vectors**

By now you must have read about vectors. A vector is a quantity
which, unlike a scalar, can have many components. For
example in Newton's second law of motion

(1) |

the quantity

In Mathematica vectors are represented in the same way.
In Mathematica this object is called a *list*, because it can
be used for more general objects such as matrices and tensors.
In this worksheet we just work with vectors.
Type ``A = Ax, Ay, Az''.
This means Mathematica associates the object A with the list
Ax, Ay, Az. Now type ``B = Bx, By, Bz''.
Type ``Dot[A,B]''.
This will give the dot product
=AxBx+AyBy+AzBz which
is the same as
, where is the angle between
the vectors and .

Likewise, the cross product of two vectors ( ) yields another vector AyBz-AzBy, AzBx-AxBz, AxBy-AyBx. ``Type Cross[A,B]'' and verify that you indeed get the above expression in terms of the components of and . Unit vectors can be easily written with lists as: 1,0,0, 0,1,0, 0,0,1. Check with Mathematica that .

You can see that the elements in the list Ax, Ay, Az of the
vector
are its *x*, *y*, and *z* components. How do we access the individual
components from A?.

Type ``A[[2]]'' and check that this gives Ay.
How would you get *Mathematica* to print out the second element
of the cross product ``Cross[A,B]''?

**Assignment 2. - Hand in by Thursday Jan. 27**

**Problem 1**. Consider two vectors
, and
. Using *Mathematica*:

(i) Check that they are both of unit magnitude.

(ii) Find
.

(iii) Find the angle between these two vectors.

(iv) Find the cross pruduct of these two vectors.

**Problem 2**.
Consider the unit vectors along x, y, and z directions:
1,0,0
0,1,0
0,0,1.
Verify:
,
,
.

**Problem 3**.
Verify that for any three vectors , , and that
.

**Problem 4**.
The motion of a particle is given by
=a(cos(t) + sin(t))
Find its velocity . Calculate
,
where
, and and verify
. Do you recognise this motion?
Plot the motion to confirm your intuition (use the
help menu to look up how to use the command ``ParametricPlot''
for this problem - you will need to choose values for
*a* and .).