I am new to Vector Spaces and I am trying to write an explan

I am new to Vector Spaces and I am trying to write an explanation of a vector space in a real world example. I had received a pretty complex and confusing definition/explanation that may have tangled me up a bit more.

I was hoping someone may be able to look over my attached image to see if I appropriately demonstrated the use of a vector space. If I have not, would you be able to appropriately advise what changes or adjustments I might need to make in order to be more in line.

I appreciate any and all help, and if you can please be thorough in your exaplanations as I am new to Vector Spaces and have been having some difficulties fully understanding them.

One real world operation that I discovered uses vectors on a daily basis are Navigational Charts for Airplanes. Each plane begins at a starting point S and after flying a set path usually in multiple distances and multiple directions arrives at its destination D. In a situation of Emergency, if the plane for whatever reason had to take a more direct course from S to D would be able to do so by calculating the distance SD using the Vectors of 100 miles South and 120 miles East. we want to find the distance from S to D and in order to do so we will apply So we see in the event of an emergency if the plane had to alter its course to arrive at its destination quicker it would save 63.79 miles by taking the more direct route. Which is a total distance of 156.21 miles instead of 220 miles.

Solution

The image and steps are very clear