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 
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 
This is a topic for another time, and comes under the term partial differential equation.…
Applications for The African Institute for Mathematical Sciences are now open
AIMS-IMAGINARY – Maths and Science Exhibition and Workshop in Mbour, Senegal
I was lucky enough to attend the first of these workshops in AIMS, Muizenberg and indeed that is where Mathemafrica was born. This should be a great event! Photo is taken from the first program:
Text below taken from the workshop/roadshow/exhibition website
This second edition of the AIMS–IMAGINARY Maths and Science Roadshow, Workshop and Exhibition will showcase interactive visual and hands-on tools used to stimulate interest in maths and sciences among diverse groups of people. The event targets primary, secondary, high school and university learners and teachers/lecturers. It will consist of (1) an AIMS–IMAGINARY Maths and Science Roadshow packed with hands-on activities and discussions, (2) an AIMS–IMAGINARY Maths and Science Exhibition, (3) discussions by interested participants willing to be part of the AIMS–IMAGINARY network to share ideas and plan for future events, and (4) a Science Slam event. The event is being organized by the African Institute for Mathematical Sciences (AIMS) and supported by IMAGINARY and the Mathematisches Forschungsinstitut Oberwolfach, and the Government of Senegal.
Game Theory can make you money…
I plan on doing some game theory posts in the near future, but for now, see how some very clever tactical thinking can do you wonders…
UCT MAM1000 lecture notes part 40 – 3D geometry and vectors part iii
The scalar, or dot product
We have seen now how to add together vectors and how to multiply them by scalars, but we haven’t seen how to multiply two vectors together. In fact it’s not all that obvious what it means to multiply two vectors together. A vector has a magnitude and a direction, how do you multiply directions? The answer is that there are two different ways to multiply together vectors. The first way which we will explore now is the scalar, or dot product. This will take two vectors and the product of them using this rule will give us a scalar. We definitely want something that is linear in both of the magnitudes of the vectors. That is to say that we want some way of multiplying together vectors so that when we double the magnitude of one of the vectors, we double the product. We will express the scalar or dot product of two vectors as: 
UCT MAM1000 lecture notes part 39 – 3D geometry and vectors part ii
Vectors
Vectors are quantities which have both magnitude (ie. size) and direction. The most common examples of these are velocity ($3ms^{-1}$ to the right) and force (10 Newtons pointing vertically down). The easiest way to describe such a quantity is an arrow, where the magnitude gives the length and the direction is given by, well, the direction of the arrow. The important point about this is that the position of the vector itself doesn’t matter. In the figure below we place the same arrow in several different places and they are all the same vector.
A vector, with magnitude given by its length and direction given by the direction of the arrow, placed at different points in the plane. Note that the position of the start of the arrow is now important, just the relationship between the start and the end of the arrow.
We could define a vector by the length and the angle that it makes with the horizontal axis, but in general we define it by how much it goes in the horizontal direction and how much it goes in the vertical direction, that is, how much it goes in the
UCT MAM1000 lecture notes part 38 – 3D geometry and vectors part i
A lot of the following is going to be rather intuitively clear, but we need to build up a framework where we are all speaking the same language to develop the powerful tools that we are going to find over the coming sections. We will be dealing here specifically with three dimensional space but we will discuss along the way the extension of these concepts to higher dimensional spaces. The higher dimensional stuff is not examinable but I think that sometimes it helps to understand the things which are special about three dimensions, and the things which are not.
In particular, I can recommend having a look at the web page of John Baez who discusses the regular polytopes in different numbers of dimensions here.
It’s clear that to define where you are in three dimensional space you need to set up a few key ingredients first. What you need is first of all an origin – a place to call home from which you will relatively describe your position.…
Galileoscope in action
Solomon Malesa, who won the translation competition has sent through photos of him in action with the Galileoscope. Congratulations again to Solomon, and thank you for the great pictures!
UCT MAM1000 lecture notes part 37 – differential equations part vi – second order differential equations
Second Order differential equations
We are only going to look at a particular subset of all possible second order differential equations (that is, equations which contain at most second derivatives) but these particular equations are absolutely ubiquitous across every field of science. The particular subset we are going to look at are linear, homogenous second order differential equations with constant coefficients. These can be written in general as:
It is linear because it contains at most (and in this case at least) a single power of 
