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By the end of this section, you will be able to:
The information presented in this section supports the following AP® learning objectives and science practices:
Suppose you want to walk from one point to another in a city with uniform square blocks, as pictured in [link] .
The straight-line path that a helicopter might fly is blocked to you as a pedestrian, and so you are forced to take a two-dimensional path, such as the one shown. You walk 14 blocks in all, 9 east followed by 5 north. What is the straight-line distance?
An old adage states that the shortest distance between two points is a straight line. The two legs of the trip and the straight-line path form a right triangle, and so the Pythagorean theorem, ${a}^{2}\text{+}{b}^{2}\text{=}{c}^{2}$ , can be used to find the straight-line distance.
The hypotenuse of the triangle is the straight-line path, and so in this case its length in units of city blocks is $\sqrt{(\text{9 blocks}{)}^{2}\text{+}(\text{5 blocks}{)}^{2}}\text{=10}\text{.}\text{3 blocks}$ , considerably shorter than the 14 blocks you walked. (Note that we are using three significant figures in the answer. Although it appears that “9” and “5” have only one significant digit, they are discrete numbers. In this case “9 blocks” is the same as “9.0 or 9.00 blocks.” We have decided to use three significant figures in the answer in order to show the result more precisely.)
The fact that the straight-line distance (10.3 blocks) in [link] is less than the total distance walked (14 blocks) is one example of a general characteristic of vectors. (Recall that vectors are quantities that have both magnitude and direction.)
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