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Displacement is defined as the final distance-vector minus the initial distance vector.

Let's say you're driving to work in the morning. You first drive north for $$5$$ miles, and then drive east for another $$5$$ miles.

Now let's say that when you get there, you take out a street map. If you were to draw a straight line starting at your house and ending at your workplace, that line would be your displacement vector. That straight line would be $$\sqrt{50}$$ miles long, (use the Pythagorean theorem) and it would be pointing northeast. Since the line has a direction, it needs to be a vector quantity. If it weren't a vector quantity, you wouldn't be able to draw it on a map at all, because it would just be a number.

Another reason why displacement needs to be a vector is that it's defined as the subtraction of two vectors, and a vector minus another vector is always a vector.

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