1.2 The Scalar Product | Geometric Algebra for Physicists

Transcript

welcome to 1.2 of geometric algebra for physicists where we will talk about the scalar product in the last video we defined a vector space and in order to arrive at a euclidean geometry we must add two new concepts to that vector space which is distance and angle the introduction of a scalar product does both of those things so let's get right into it we're starting with the scalar product properties so given two vectors a scalar product is a rule for obtaining a number with the following properties which again these are very basic properties no proof needed it is uh interesting to keep note that addition is symmetric and not anti-symmetric given the formation properties and vector a if you take the absolute value of a then you get the distance this is also known as the length of vector a or the magnitude of vector a and there are plenty of different names but i'm mostly just going to either use the word length or magnitude note that the introduction of distance makes this vector space a metric space and this in turn creates a euclidean space whose inner product is positive definite meaning the schwarz inequality is of the following form and now there is a proof right here uh it's a non-rigorous proof taken from the textbook and the only part that maybe might not look super intuitive is like this step to this step but this is if you take the quadratic discriminant which if you remember that's a b squared minus 4ac and so this is b squared and then 4ac right here but you just like transfer it over and then it simplifies to this and you might be wondering like how how did you get this uh absolute value science right here well the reason is because that it's a positive definite space now and because of that then even if you didn't have these lines right here eight dot b vector will still be equal to the magnitude of a dot b vector because the scalar is positive definite now we also have angle between vectors and one can find that through the following equation which is a dot b is equal to the magnitudes or lengths of a and b times cosine theta now a really interesting tool to visualize this is let's just say that you have angle or vector a right here and then vector b is some vector like this well what this equation is doing is basically overlaying this vector it puts it basically puts a vector parallel to a which is basically the component of b that goes in the direction of a so it's basically saying that how how much of b goes in the direction of a like how much of b goes in the direction of a and it's pretty easy to visualize right here if you want a little bit more of a mathematical proof of it you just have theta here and this is b well right here this length right here using basic trig is cosine theta times the magnitude of b right there and so you have this equation right here which is a times the magnitude of b right here which is going to basically give you the length that is for right here and a and b are orthogonal vectors if and only if their inner if the their product the scalar product is equal to zero now it's easy to think about this like if these are perpendicular right here then it's pretty easy to visualize that this will be zero because if you have 90 degrees right here cosine of 90 degrees which is also a cosine of pi over 2 that is equal to zero if you think about on the unit circle as you go there that becomes zero and so it's pretty easy to see that a dot b vector is equal to zero now for the definition of orthonormal if all basis vectors are mutually orthogonal and are normalized to unit length this basis is called orthonormal for example if the set e to e sub n vectors denote such a basis the statement that the basis is orthonormal can also be summarized as e sub i dot e sub j vectors is equal to the delta function which is also known as the kronecker delta function which recurs uh pretty often in a linear algebra intense or calculus where it has the following properties at if i equals j then the function is equal to one and if i doesn't equal j then the chunks is equal to zero if you think about a matrix with the diagonals basically um it's saying that any vectors with products with who are off diagonal will be equal to zero and only diagonals will be there an interesting thing is that if you make a matrix out of the kronecker delta function in any dimension that creates the i that creates an identity matrix which is just the matrix times that is equal to the original matrix and you can expand any vector a in this basis as this sum right here which actually this right here is what we're going to get to next which is einstein's summation convention which is extremely useful in tensor calculus and linear algebra because it gets rid of this mess right here granted it's not a mess and it's very it's very precise but these indices become assumed right here and you only need to represent it using the variables and so this sum right here just becomes the single variables next to each other which as you can see they still represent the exact same thing now this comes and shows something really useful in vector component representation you can see the definition basically right here saying that you get the you get this scalar here by dot producting a unit vector and a basically that creates this following equation right here a dot b is equal to using these substitutions of vector component representation it simplifies to this and then notice that you substitute in the kronecker delta function which just makes it basically so b only will have i components because otherwise that'll be equal to zero so it cancels out and so it creates a very simple a representation of a vector in scalar products it only becomes this which is just the sum of a sub 1 times b sub 1 the components of the respective vectors plus going to infinity a sub n to b sub n so yeah thank you for watching next lesson will be on complex numbers and basically how to interpret them in a geometric algebra perspective it will be relatively simple again because this chapter in the textbook is pretty much only review and getting you used to the ideas before you jump right into geometric algebra which will start being covered in chapter 2. so yeah thank you