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# Can displacement be defined in more than one dimension?

More often than not you have to define a displacement in two or three dimensions. As an example, suppose you want to locate a house that is 160 feet west of 1st Avenue and 200 feet north of Main Street. The displacement is a combination of 160 feet west and 200 feet north. But how are they combined? You can’t simply add them, because they have different directions. Go back to the drawing with the arrow. Define north as the direction toward the top of the page. Then add a second arrow 2.0” long in the upward direction. The tails of the two arrows are at the same place, representing the intersection of Main Street and 1st Avenue.

The two arrows are half of a rectangle 1.6” wide and 2.0” high. Draw lines completing the rectangle. The location of the house would be at the upper right-hand corner of the rectangle. Draw a third arrow, with the tail at the intersection of the other two vectors and the heat at the upper right-hand corner. The length of the arrow can be measured on your drawing, or calculated using the Pythagorean Theorem: the square of the length (the hypotenuse of a right triangle) is equal to the sum of the squares of the other two sides. In this case: 1.62 2.02 = 6.56. Then length is the square root of that, or 2.56”. So in real life the displacement would have a magnitude of 256 feet.

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