Distance And Displacement

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A cyclist rides 600 m due east, then turns and rides 800 m due north before stopping at a junction. What is the magnitude of the cyclist's total displacement?

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Displacement is a vector quantity defined as the straight-line change in position from start to finish, whereas distance is the total scalar path length actually traversed. Because the two legs of the ride are along mutually perpendicular directions (east and north), the resultant displacement is the vector sum of two orthogonal vectors and is found using the Pythagorean theorem. Taking east as the $x$-axis and north as the $y$-axis, the net position vector is $(600\,\hat{i} + 800\,\hat{j})	ext{ m}$. Its magnitude is $\sqrt{600^2 + 800^2} = \sqrt{360000 + 640000} = \sqrt{1000000} = 1000	ext{ m}$. The option 1400 m is wrong because it simply adds the two legs, which gives the total distance, not the displacement. The option 200 m is wrong because subtraction has no physical basis for perpendicular legs. The option 700 m is an arbitrary intermediate value with no derivation. This is the standard NCERT treatment of vector addition of perpendicular displacements. As a sanity check, the displacement (1000 m) is correctly smaller than the path length (1400 m), which must always hold whenever the motion is not perfectly straight.

Distance And Displacement

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