 So the take-home message is there is no special place that has to be the origin. Alice can say the origin is over here, Bob can say the origin is over there, and both of them are equally valid. We call these two different reference frames, and if I want to say the treasure is buried at six and a half comma zero kilometers, I have to say relative to Alice's frame or relative to Bob's frame, and this is where the relative and relativity comes from. But not everything is relative. So let's suppose this is Alice's reference frame where we use xa and ya to mean the x and y axis of Alice, and Bob is standing at one zero relative to Alice. Bob throws a ball and it lands at four four relative to Alice. Now let's look at things from Bob's frame. Now Bob's frame is rotated with regards to Alice. Now this is all fine, just like there's no special point that we have to call the origin. There's no special direction that we all have degrees the x and y axis allow. You might say why can't everyone say the north pole is the y axis, but someone on Mars might disagree. So there is no special orientation, and Bob is just as well looking at his rights to say I'm the straight one and Alice is rotated relative to me. So if they want to figure out how far the ball went, they can just measure the distance in their reference frame. Now in Bob's frame it's easy. It started off at zero zero and landed at five zero, which gives us a distance of five. In Alice's frame we have to do a bit of trig, but that's fine, and we find the distance is still five. So even though points change, for example Bob was at one comma zero in Alice's frame and zero comma zero in his own frame, the distance between two points calculated using Pythagoras is what we call an invariant. It is the same in anyone's reference frame. Provided that the two frames differ from each other either by a translation, they have the origin in different places, or by a rotation, the axes are pointing in different directions. Lastly, notice how easy it was to move from describing things in Alice's reference frame to describing things in Bob's reference frame. So even if the two frames differ from each other by a translation or a rotation, we can always transform from one reference frame to the other.