hihi! could use some help with understanding position vectors- just seeing it in questions confuses me :/
Hey there! I must be a year late or something by now but hopefully this can still be helpful for someone out there. A position vector is what it sounds like: a vector describing position. Now the word vector has many meanings in many different fields. Theyâre all very similar and some would argue theyâre all just different ways of saying the same thing, but in this context what we mean when we say a vector is basically an ordered list of numbers: (1,2,3) or (5, -10, 12), or (1,5). A vector can have as many entries as you want it to, but for our purposes letâs focus on just vectors of length two right now -- that is to say, vectors with two entries. The idea of a position vector is to use every entry of the vector to describe some aspect of the position of something, relative to another point (usually the origin). Imagine you live on a 2D plane. You want to get to your friendâs house and you know itâs 4m east and 3m north of your house. Relative to your home then, you can describe the position of your friendâs house as (4, 3). Here the first entry describes how far east they are from you and the second entry is for how far north. If thereâs a store 5m west and 2m south of your house you could describe the position as (-5, -2) where again the first entry is how far âeastâ they are and the second is how far ânorthâ (the numbers are negative since the store is in the opposite direction of east/north) Here you can see the first entry is really describing the horizontal position relative to your house and the second entry is vertical position. Putting these two coordinates together gives you the unique position of the location. Notice, thereâs no law saying the first entry HAS to be the horizontal position. You can flip them around if you want as long as you know what you mean and youâre consistent with it. Normally though, people use them the way described for convenience because then you can assume everyone is on the same page. The real power of position vectors comes from adding or subtracting them. Letâs think about what that means. If you imagine your house and your friendâs house on the plane, then you can think of the vector we talked about earlier (4, 3) as an arrow pointing from your house to theirs. This is great if you know the shortest path to their house, but if this is your first time visiting them you might not. Maybe you have a cryptic list of instructions from them which says: 1. Go 2m west and 4m north to the park 2. Then go 6m east and 2m south to the supermarket 3. Finally go 1m north to get to my house. You know then that relative to you, the park is at position (-2, 4). Relative to the park the supermarket is at position (6, -2) Finally, relative to the supermarket their house is at position (0, 1) In the same way that we imagined the position vector of their house relative to ours as an arrow pointing from us to them, we can think of each of these position vectors in a similar way. Intuitively, we know then that to get to their home we just have to follow the arrows from start to end. After all the steps weâll arrive at their house. The idea of âfollowing the stepsâ is simply the idea of adding the vectors, where you add them by adding the entries in the same positions together: (for example (1, 2) + (3, 4) = (1 + 3, 2 + 4) = (4, 6)) Therefore we know we can add the vectors for each step of the route to our friendâs house, and it should therefore just give us the position of their house relative to ours. Doing so yields: (-2, 4) + (6, -2) + (0, 1) = (-2 + 6 + 0, 4 + -2 + 1) = (4, 3) I appreciate this is a bit rushed and compacted so there may be parts that could benefit from further explanation, but regardless I hope that this is helpful for anyone who might read it.Â
