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If I have a vector sitting here in 2D space we have a standard way to describe it with coordinates. In this case, the vector has coordinates [3, 2], which means going from its tail to its tip involves moving 3 units to the right and 2 units up. Now, the more linear-algebra-oriented way to describe coordinates is to think of each of these numbers as a scalar a thing that stretches or squishes vectors. You think of that first coordinate as scaling i-hat the vector with length 1, pointing to the right while the second coordinate scales j-hat the vector with length 1, pointing straight up. The tip to tail sum of those two scaled vectors is what the coordinates are meant to describe. You can think of these two special vectors as encapsulating all of the implicit assumptions of our coordinate system. The fact that the first number indicates rightward motion that the second one indicates upward motion exactly how far unit of distances. All of that is tied up in the choice of i-hat and j-hat a
