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:

int

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.…

By | September 26th, 2015|Courses, English, First year, Level: intermediate, MAM1000, Uncategorized, Undergraduate|4 Comments
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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.…

By | September 25th, 2015|Courses, English, First year, Level: intermediate, MAM1000, Uncategorized, Undergraduate|3 Comments
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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.…

By | September 20th, 2015|Courses, English, First year, Level: intermediate, MAM1000, Undergraduate|0 Comments
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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.…

By | September 19th, 2015|Courses, English, First year, Level: intermediate, MAM1000, Uncategorized, Undergraduate|6 Comments
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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.…

By | September 19th, 2015|Courses, English, First year, Level: intermediate, MAM1000, Undergraduate|1 Comment
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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}.…

By | September 15th, 2015|Courses, First year, MAM1000, Uncategorized, Undergraduate|1 Comment
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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: \vec{v}.\vec{w}.…

By | September 5th, 2015|Courses, First year, MAM1000, Uncategorized, Undergraduate|3 Comments
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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.

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 x-direction and how much it goes in the y-direction.…

By | September 4th, 2015|Courses, First year, MAM1000, Uncategorized, Undergraduate|1 Comment
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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.…

By | September 4th, 2015|Courses, English, First year, Level: Simple, MAM1000, Uncategorized, Undergraduate|1 Comment
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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:

 

\frac{d^2y}{dx^2}+b\frac{dy}{dx}+c y=0

 

It is linear because it contains at most (and in this case at least) a single power of y in each term. It is homogenous because there is no term which has no powers of y (ie. the right hand side is not a constant), and the coefficients b and c are any real numbers (though you can extend this to having complex numbers very easily). We will see that depending on the relationship between these numbers (b and c) we can have very different behaviour of the equation.…

By | September 1st, 2015|Courses, English, First year, Level: intermediate, MAM1000, Uncategorized, Undergraduate|3 Comments
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