What significance is the product of two elements within the same dimension?

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Say there are two vectors A (1, 0) and B (2, 0) written as cartesian coordinates (x, y). What type of result does multiplying the x elements together produce? By this I mean multiplying 1 and 2 to get result product 2. Does this resulting product 2 have any real world interpretation? If x and y are considered "dimensions", does the resulting product 2 live in x as well or another dimension relative to it? Does it have any units? e.g. if x has units in meters, does the product 1 meter * 2 meter give us 2 meters squared?