This is not how I would have thought to prove why the polar formula for dot product doesn't get any more complicated in higher dimensions, while the "sum of the products" formula does, but I also would not have noticed the connection to the law of cosines without this approach, so that's a new insight for me. As an aspiring math teacher, I also really like the effort made to get the students physically involved with pens as vectors and the book as a plane. It demonstrates what I would think is a very simple idea to imagine, but I also know that it's easy to underestimate the value of a literal hands-on representation. There's value in that simple demonstration, that is easy for a math teacher to miss.
What a great explanation of why we can apply the formula for the dot product (that is derived from the law of cosine which is only applicable to the 2 dimensional figure that is the triangle) in 3 dimensions. I actually just had the epiphany right before you had that demonstration. I guess the way I explained it to myself is that even though we are in three dimensions(or perhaps even more), we are still working with triangles, not pyramids.
Funny thing, I used this + Euclidian Distance to get a really good initial population for genetic algorithm for solving Vehicle Routing Problem, paper will be out soon.
I use this in my statistics class to prove that Pearson's Correlation coefficient, r, is bounded by -1 and 1 as the formula for correlation is sum[(x-mean(x))*(y-mean(y))]/[magnitude(x-mean(x))*magnitude(y-mean(y))] which is cos theta! Theta is the angle between the (x - mean(x)) and (y - mean(y)) vectors.
Yes! the two vectors will always be coplanar no matter how many dimensions we are considering. Just like how we can create 2dimensional slices in 3D, we can do the same in 4D. the only thing affected is how that slice is orientated in each dimension.
I love maths alot.i have passion for maths sir.i want to be a professor in maths.That what prompted me to do video on UA-cam I will be grateful if you can post me sir or add me to your channel
This is not how I would have thought to prove why the polar formula for dot product doesn't get any more complicated in higher dimensions, while the "sum of the products" formula does, but I also would not have noticed the connection to the law of cosines without this approach, so that's a new insight for me.
As an aspiring math teacher, I also really like the effort made to get the students physically involved with pens as vectors and the book as a plane. It demonstrates what I would think is a very simple idea to imagine, but I also know that it's easy to underestimate the value of a literal hands-on representation. There's value in that simple demonstration, that is easy for a math teacher to miss.
What a great explanation of why we can apply the formula for the dot product (that is derived from the law of cosine which is only applicable to the 2 dimensional figure that is the triangle) in 3 dimensions. I actually just had the epiphany right before you had that demonstration. I guess the way I explained it to myself is that even though we are in three dimensions(or perhaps even more), we are still working with triangles, not pyramids.
Watching this guys 2014 vids on zero index. He been smart for a Wwhiillleee
Funny thing, I used this + Euclidian Distance to get a really good initial population for genetic algorithm for solving Vehicle Routing Problem, paper will be out soon.
I use this in my statistics class to prove that Pearson's Correlation coefficient, r, is bounded by -1 and 1 as the formula for correlation is sum[(x-mean(x))*(y-mean(y))]/[magnitude(x-mean(x))*magnitude(y-mean(y))] which is cos theta! Theta is the angle between the (x - mean(x)) and (y - mean(y)) vectors.
Really wish he was my school teacher
yeah
Would this formula work with vectors in 4 spatial dimensions?
Dot product works for all dimensions (as long as the space itself is linear).
Yes! the two vectors will always be coplanar no matter how many dimensions we are considering. Just like how we can create 2dimensional slices in 3D, we can do the same in 4D. the only thing affected is how that slice is orientated in each dimension.
@@fivenightsofben6096 does this mean 3d planes in 4d space?
He made it so easy
I love maths alot.i have passion for maths sir.i want to be a professor in maths.That what prompted me to do video on UA-cam I will be grateful if you can post me sir or add me to your channel
0rd