MATH6040: Week 12

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

Test 2

You will get a chance to view your paper in the tutorials of Week 13.

Academic Learning Centre Questions

P.163 & 171 the Q. 3s can be done together.

Week 12

We finished off the course by talking about centroids of laminas and centres of gravity of solids of revolution. There are marking schemes to Q.4 questions on pages 158, 168 and page 179. I have also done out some more examples of questions for 4 (c) on centroids and centres of gravity — I will give these to you in Week 13.

Week 13

There is an exam paper at the back of your notes — I will go through this on the board in the lecture slots:

Tuesday 11:00 and 15:00
Thursday 11:00

In the normal rooms.

Note you won’t get a question like Q.3 (c) but I have a replacement question done out.

We will also have tutorial time in the tutorial slots. You can come to any or both tutorials.

Monday at 14:00
Thursday at 14:00

Again in the normal rooms.

Study

Please feel free to ask me questions about the exercises via email or even better on this webpage.

Student Resources

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc..

4 comments

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May 16, 2015 at 2:59 pm

Student

Hi J.P.,

I’ve just a quick question on unit vectors.

Is it just $\mathbf{v}$ over it’s own magnitude, $|\mathbf{v}|$ ?? Do you have have to work it out or just leave it like that?

Thanks.

May 16, 2015 at 3:09 pm

J.P. McCarthy

Let us say we have a vector $\mathbf{v}$ . Now there are two equivalent notations for a vector in space:

$\mathbf{v}=a\mathbf{\hat{i}}+b\mathbf{\hat{j}}+c\mathbf{\hat{k}}$

$\mathbf{v}=(a,b,c)$ .

Consider these examples.

Example 1:
Find a unit vector in the same direction of $\mathbf{v}=2\mathbf{\hat{i}}+3\mathbf{\hat{j}}-6\mathbf{\hat{k}}$ .

Solution: We find

$|\mathbf{v}|=\sqrt{(2)^2+(3)^2+(-6)^2}=\sqrt{49}=7$ .

Now as this is such a nice number it is very easy to do write:

$\displaystyle\hat{\mathbf{v}}=\frac{\mathbf{v}}{|\mathbf{v}|}=\frac{(2\mathbf{\hat{i}}+3\mathbf{\hat{j}}-6\mathbf{\hat{k}})}{7}$

$\displaystyle=\frac27\mathbf{\hat{i}}+\frac37\mathbf{\hat{j}}-\frac67\mathbf{\hat{k}}$

Example 2:
Find a unit vector in the same direction of $\mathbf{u}=(2,-1,0)$ .

$|\mathbf{u}|=\sqrt{(2)^2+(-1)^2+(0)^2}=\sqrt{5}$ .

Now as this is NOT such a nice number it is easier to write:

$\displaystyle\hat{\mathbf{u}}=\frac{\mathbf{u}}{|\mathbf{u}|}=\frac{(2,-1,0)}{\sqrt{5}}$

which is fine.

In other words just write

$\displaystyle\hat{\mathbf{v}}=\frac{\mathbf{v}}{|\mathbf{v}|}$ and I will give you the marks but it would be good to divide in if the number, $|\mathbf{v}|$ , is nice.

Regards,
J.P.

May 17, 2015 at 3:18 pm

J.P.

Can you send Q 1 (b) please: I totally forget.

May 17, 2015 at 3:23 pm

Let $x$ be the first number and $y$ be the second.

The first condition gives

$x+y=9$

and the second gives

$2x+3y=20$

so the simultaneous equations are

$x+y=9$
$2x+3y=20$

To use the matrix inversion method you must write this as a matrix equation:

$\left(\begin{array}{cc}1 & 1 \\ 2 & 3 \end{array}\right)\left(\begin{array}{c} x \\ y\end{array}\right)=\left(\begin{array}{c} 9 \\ 20\end{array}\right)$

Note this has the form

$A\mathbf{x}=\mathbf{b}$ .

To solve this we left-multiply both sides by $A^{-1}$ :

$A^{-1}A\mathbf{x}=A^{-1}\mathbf{b}\Rightarrow \mathbf{x}=A^{-1}\mathbf{b}$

There is a formula for the inverse of a $2\times 2$ matrix in the tables:

$\left(\begin{array}{cc}a & b \\ c & d\end{array}\right)^{-1}=\frac{1}{ad-bc}\left(\begin{array}{cc}d & -b \\ -c & a\end{array}\right)$

I trust you can finish this off?

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MATH6040: Week 12

Test 2

Academic Learning Centre Questions

Week 12

Week 13

Study

Student Resources

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J.P. McCarthy Maths

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MATH6040: Week 12

Test 2

Academic Learning Centre Questions

Week 12

Week 13

Study

Student Resources

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Recent Comments

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J.P. McCarthy Maths

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