For those who don’t know, a huge amount of Khan academy has been translated into Xhosa. The videos can be found here.

For instance, here’s the binomial theorem:

as translated by Zwelithini Mxhego. It looks like these videos haven’t been viewed all that many times, though a huge amount of work has clearly gone into them. If you think you know people who might benefit from these, please do spread the word!

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## UCT MAM1000 lecture notes part 45 – 3D geometry and vectors part viii

We will discuss mostly three dimensions here, but what we have will be applicable to any number of dimensions (greater than or equal to 1). We want to be able to describe a straight line – a one dimensional object, infinitely long in both directions. We will see that vectors give us a perfect language with which to do this.

Remember that in three dimensions, a line can be defined by the intersection of two planes as in the intersection of the blue and the green planes defining the red line:

Each plane is specified by a single equation, and thus a line is specified by two equations (one for each plane). Here we will see that sometimes you just need one equation to specify a line, if you are using vectors, and sometimes it will seem that you need three equations, if you are using a parametric equation.

Let’s take a line, and specify some point on it.…

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## UCT MAM1000 lecture notes part 44 – 3D geometry and vectors part vii

In the following, I’m going to miss out quite a few details which I think are very nicely laid out in Stewart. I will try and add a slightly more pedagogical tone to some of it, and some nice diagrams along the way.

So we saw in the last post that we can write the cross product of two vectors, which itself gives a vector, in terms of the determinant of a 3 by 3 array. We can use this to both find a vector perpendicular to two given vectors (unless they are parallel to one another) and also to find the area of a parallelogram formed by two vectors (the area of which is zero if the vectors are parallel to one another).

The second of these is easy enough to do in two dimensions, but in three dimensions that’s not an easy prospect. Using the cross  (otherwise called the vector) product makes this easy.…

## UCT MAM1000 lecture notes: More complex numbers practice

I’ve been asked a few times for more practice questions on complex numbers. This is where Wolfram Alpha can be your friend (like it’s not already!).

I’ll just give a few examples of questions from the tut on complex numbers which you could have solved using Wolfram Alpha, and from this you will be able to set up your own questions.

For instance, question 48 c) Find the roots of $z^5=1+\sqrt(3) I$ can be solved in Wolfram Alpha with the command:

Solve[z^5==1+sqrt[3]I,z]

Moreover it will solve this for you, give you the five roots and plot them in the complex plane. So now you can come up with any root question you can possibly think of. There’s an infinite number of questions to start you off. You can thank me later!

If you want to convert between the trigonometric form and the exponential form, you can use the two commands:

TrigToExp[Sin[x]+2 I Cos[x]]

ExpToTrig[Exp[I z+3]]

Though remember the definition of the hypergeometric trig functions from a previous tut.…

## UCT MAM1000 lecture notes part 43 – 3D geometry and vectors part vi

Determinants

The idea of determinants have been about since around the 3rd century when it first appeared in an ancient Chinese book of Mathematics called The Nine Chapters on the Mathematical Art. It was used originally to define certain properties of systems of linear equations, as we will see later in the section on linear algebra, however for now we will simply use it as a particular way to easily calculate the cross product. Let’s take a two by two array of numbers and define the determinant for this.

$\left|\begin{array}{cc}a & b \\ c & d \\\end{array}\right|=ad-bc$

The vertical lines on the left and right are the sign that the we are taking a determinant. For now this is just a definition and we will work with it in what follows. Don’t worry too much about where it comes from, but we will see later where it comes from and we will see now why it is useful.…

## UCT MAM1000 lecture notes part 42 – 3D geometry and vectors part v

The vector, or cross product

When we took two vectors previously and found a way to multiply them together using the dot product, we ended up with a scalar. However, there is also a way that we can take two vectors and multiply them together to give a vector, but a vector with very specific properties with respect to the first two. What we will define here will be in three dimensions, and, unlike the dot product, does not generalise easily to other dimensions, (other than 7) though it can in fact be extended.

We are going to define the cross product such that it gives a vector which is perpendicular to the two vectors being crossed. This might sound a bit arbitrary but it shows up in a huge number of different situations in physics in particular and can help us to understand the geometric relation between vectors very simply.…

## Something for the weekend: Pascal’s Triangle at TED

Pascal’s triangle is even deeper than you thought…

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## UCT MAM1000 lecture notes part 41 – 3D geometry and vectors part iv

Scalar and vector projections

Given two vectors, can we ask how much of one vector is pointing in the direction of the other? We can certainly ask how much of the vector $\left<5,6\right>$ is pointing in the $x$ direction – the answer is just 5. You can think of this as projecting the vector onto the $x$-axis and asking for its projected length. Similarly we can ask about the projection of a vector into any arbitrary direction. This is illustrated in figure \ref{vec6}. Imagine having a light perpendicular to $\vec{b}$ shining towards it. There is a shadow of the vector $\vec{a}$ cast on the line of $\vec{b}$. This is the scalar projection of $\vec{a}$ in the direction of $\vec{b}$, also called the component of $\vec{a}$ in the direction of $\vec{b}$. When you are looking at this, clearly the size of $\vec{b}$ is unimportant, so you can think of an infinite line stretching in both directions parallel to $\vec{b}$.…

## Chaos from differential equations

In all of this talk about differential equations, we haven’t spoken all that much about the uses of them, apart from a little about population dynamics, nor indeed about their amazing properties. Part of the reason for this is that in general (though of course not exclusively), the most interesting differential equations are a single step beyond what we have been looking at. They are differential equations in more than one variable. For instance, rather than just having a $y$ be a function of $x$ or $t$, they have $y$ a function of both $x$ and $t$. It turns out that this little change makes all the difference in the world. All of a sudden we can see how things change in both space and time. We can look at real dynamics of systems which are not local to a single place.

This is a topic for another time, and comes under the term partial differential equation.…