Geometry

Geometry is a field of mathematics concerned with spacial objects. It provides ways of formally quantifying intuitive concepts such as distance, shape and size.

Table of contents

    Intro

    Geometry is on of the oldest area of mathematics. It is easy to imagine humans scribbling geometric shapes in the sand while we roamed the earth in ancient times.

    Geometry is all about shapes, distances position, sizes, etc. It is our way of making sense of the natural world. Geometry literally means "earth measuring".

    There are countless applications of geometry. For example, in computer graphics, CAD-programs, civil engineering etc.

    Concept

    Back in boring old single variable calculus, we mostly dealt with curves and lines in two dimensions with the -plane.

    In multivariable calculus where we introduce the new variable , the old curves and lines becomes planes and surfaces in the -space.

    A few examples are

    Math

    In math and physics we often deal with vectors. Vectors as you know is like a list of numbers. We can add vector and subtract vector, we can also multiply vectors with a number. But there are two more operator we can to with vectors. These two are called the scalar and cross product.

    The scalar product "" between two vectors in is calculated like below

    The cross product is calculated as follow

    The scalar and cross product have very geometric meaning that will be investigated in detail in the lecture notes.

    Quadratic surfaces

    Studying multivariable calc, you'll have to draw lots of funky 3D shapes. Add some colors here and there, and your math notes will be really pretty. You can flaunt your notes on your family's Christmas gathering, telling grandma how neat math is.

    In multivariable calc, you'll run into equations like

    You might be able to factor the whole thing, like

    in which case the graph consists of a pair of planes.

    But let's assume that ain't the case. It's more fun.

    If , you've got a cylinder. This is illustrated by graph to the left below.

    You could have , like in the graph to the right. This guy also belongs to the cylinder-family, and it's called a parabolic cylinder.

    Equations like results in a cone. See the left hand side graph below. If we set , we end up with a sphere, like in the graph to the right.

    Let's move on to the paraboloid.

    There are two versions, which look a bit different from one another. First, let's say . Then you'll get the thingy to the left.

    It may also be the case that . Then, the surface looks more like in the picture to the right.

    Let's end our journey through the land of 3D shapes with the hyperboloid.

    There are two cases: first the positive one, for example , which you can see to the left below.

    Then, to the right you have the negative case .

    Cross product

    Multiplication between two vectors is not defined, but there are two definitions where multiplication is still used between the elements: dot product (or scalar product) and cross product (or vector product).

    The cross product of two vectors is a third vector, and it will form angles to the other two

    While the dot product of two vectors results in a scalar, the cross product between two vectors is a new vector.

    The vector resulting from a cross product will be perpendicular to the two vectors being multiplied. This means it forms a angle to both of them.

    Another difference between the two types of vector multiplication is that the cross product is only defined for three dimensions (), while the dot product is defined for all dimensions of space.

    Let us examine how the cross product of two vectors is computed:

    Let and be vectors in . Then the cross product of and is noted as and is defined as:

    The following properties apply to the cross product:

    Let , and be vectors in and be a scalar. Then we have:

    Vector projections

    So you're out for a walk, and you grab a stick from the ground. It has a "V" shape, like this:

    It's a sunny day, and the upper part casts a shadow on the lower twig. Since you're kind of geeky, you try to compute the length of the shadow. This is, of course, a problem of uttermost importance!

    Now we turn to linear algebra. Let's call the upper twig and the lower one . The tip of might lie at and the tip of at . What to do? Well, we've got a handy lil' formula for that. Just compute

    That's your shadow, more formally referred to as a vector projection. Now the little dot between and signifies one of the two ways in which vectors can be multiplied together. Yep, you guessed it: The dot product.

    Dot product

    The dot product results in a scalar (a number), and is therefore also known as the scalar product. Usually, it is denoted as either or .

    Let and be two vectors in :

    then the dot product is algebraically defined as:

    and the dot product has the following geometric definition:

    where refers to the length of the vector and refers to the angle between and .

    The scalar product is also defined for with the same geometric relationship as above for . The calculation works analogously:

    Consequently, the shadow that casts on is:

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