Components and Coordinates

The topic is muddled because if you read enough web pages you find people switching the words and this might make you think they are perfect synonyms. ChatGPT insists that they are not and it expounds on this to make the word component take you to one story and the word coordinate to take you to another. Both stories have visual representations which are shown in the picture.

In the story on the left we show the idea of the components of a vector. We can argue that we are working free of a coordinate system, we are simply showing…

…and in this story and are components. Note carefully that we are requiring that a component have both a number and a “hat”. Take a deep breath. It’s not hard to imagine that you will find writer after writing telling you that the component is just the number. (See Appendix A for Dissenting Opinions)

In the story on the right we show the idea of a point and its coordinates. The point is (4,3) and we find it on the “grid”. Now, when you stare at the grid you will probably notice that it is not straight lines, but rather, collections of copies of a basis vector added one to the other to stretch out in a straight line. Take note that the point (4,3) is the intersection of the x=4 coordinate with the y=3 coordinate.

To better understand coordinates, look at a piece of graph paper with blue lines on it–usually the blue lines represent integers (x=1, x=2, x=3, etc.) Notice that on a 2D graph those coordinates are vertical lines while the coordinate axis that we call the “x-axis” is horizontal. Something similar happens for y, with coordinates being horizontal and the coordinate axis we call the “y-axis” being vertical. You can see how this could create confusion if someone answers a question with an incomplete explanation.

The last part of this discussion focuses on getting from one to the other. You noticed that we built the grid using information that was available in the story about components. Yes, once you have the basis vectors from the component story you can start putting them together to make the lines for a grid and those lines will be coordinate lines.

The above is a lot of discussion. We leave with two summary points:

• Coordinates specify position and a coordinate is just a number.
• Components represent projections on coordinate axes and a component is a geometric vector.

Appendix A

It gets tricky. In math you are allowed to define the things you are using and then you must be self-consistent within your own system. This lets people like Buck (textbook author, his work was awesome) tell us that we can add points and then later someone tells us that we don’t do that. We are restricted (in physics) using only vectors which are geometric vectors because we want a vector to have a magnitude and a direction.

We are telling you that a vector component looks like and then warning you that the word might be used for just the number.