Worksheet #2 - PHY102 (Spr. 2005)
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

$$\vec{F}=m\frac{d^2\vec{r}}{dt^2}$$ (1)

the quantity m (mass) is a scalar. But the force $\vec{F}$ and the acceleration $\vec{a}=\frac{d^2\vec{r}}{dt^2}$ are vectors. As you can see in Eq. (1), and which is true in general, multiplying a vector $\vec{a}$ with a scalar m, gives a vector $\vec{F}$. A vector is described by its components in a chosen co-ordinate system. For example a vector $\vec{A}$ in cartesion co-ordinates is given by, $\vec{A}=(A_x,A_y,A_z)$.

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 $\vec{A}\cdot{\vec{B}}$=AxBx+AyBy+AzBz which is the same as $\vert A\vert\vert B\vert cos(\theta)$, where $\theta$ is the angle between the vectors $\vec{A}$ and $\vec{B}$.

Likewise, the cross product of two vectors ( $\vec{A}\times\vec{B}$) yields another vector $\vec{C} = \{$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 $\vec{A}$ and $\vec{B}$. Unit vectors can be easily written with lists as: $\hat{x} = \{$1,0,0$\}$, $\hat{y} = \{$0,1,0$\}$, $\hat{z} = \{$0,0,1$\}$. Check with Mathematica that $\hat{x}\cdot\hat{y} = \hat{y}\cdot\hat{z} = \hat{z}\cdot\hat{x}= 0$.

You can see that the elements in the list $\{$Ax, Ay, Az$\}$ of the vector $\vec{A} = \hat{x}A_x + \hat{y}A_y + \hat{z}A_z$ 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 $\vec{A} = (\frac{\sqrt{3}}{2},\frac{1}{2},0)$, and $\vec{B}= (\frac{1}{2},\frac{\sqrt{3}}{2},0)$. Using Mathematica:
(i) Check that they are both of unit magnitude.
(ii) Find $\vec{A}\cdot{\vec{B}}$.
(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: $\hat{x} = \{$1,0,0$\}$ $\hat{y} = \{$0,1,0$\}$ $\hat{z} = \{$0,0,1$\}$. Verify: $\hat{x}\times\hat{y}=\hat{z}$, $\hat{y}\times\hat{z}=\hat{x}$, $\hat{z}\times\hat{x}=\hat{y}$.

Problem 3. Verify that for any three vectors $\vec{A}$, $\vec{B}$, and $\vec{C}$ that $\vec{A}\times(\vec{B}\times\vec{C})= \vec{B}(\vec{A}\cdot\vec{C})-\vec{C}(\vec{A}\cdot\vec{B})$.

Problem 4. The motion of a particle is given by $\vec{r}(t)$=a($\hat{x}$cos(${\omega}$t) + $\hat{y}$sin(${\omega}$t)) Find its velocity $\vec{v}$. Calculate ${\vec{\Omega}} \times \vec{r}$, where $\vec{\Omega}=(0,0,\omega)$, and and verify $\vec{v}={\vec{\Omega}} \times \vec{r}$. 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 ${\omega}$.).