How does a vector work in the fourth dimension?
In: Physics
same as they do in 2D and 28D. Dimensions with vectors aren’t really what we think of them in day to day, they are significantly more abstracted and don’t need to be location in space. You can have a vector of prices, or voltages, or even both of those things.
But the math works the same way. As an example, let’s look at a basic operation called the dot product. Let’s start with 2 2D vectors [x1 x2] and [y1 y2]. The dot product is x1y1+x2y2. Pretty simple right each of the elements are multiplied by the corresponding one in the other vector and then everything is added. Now lets look at 2 4D vectors [x1 x2 x3 x4] and [y1 y2 y3 y4]. The dot product is x1y1+x2y2+x3y3+x4y4. Same pattern. The only issue is that you can’t graph them or really even look at them, you just have to rely on what the math says.
Vectors work the same in a 4 dimensional space as they do in a 2 dimensional space. Sure, you can’t visualize them, because you don’t have 4 dimensional space to work in, but all the math operations work the same way.
Instead of imagining dimensions as purely axes of Cartesian space, imagine them as simply a quantifiable measure of something. It could be intensity (of light for example), XYZ position, time, color, number of sneezes, whatever. Now a vector can be imagined as a change of some quality (or qualities) of an object in the reference frame of some combination of dimensions. That’s it.
Latest Answers