A person desires to reach a point that isfrom her present location and in a direction that isnorth of east. However, she must travel along streets that are oriented either north–south or east–west. What is the minimum distance she could travel to reach her destination?
The minimum distance traveled to reach destination is
This problem refers to scalar projection. In Cartesian coordinates, scalar components are scalar projections in the directions of the coordinate axes. Using this concept, the distance can be calculated by finding the components and adding those components.
The components can be written as
Therefore the minimum distance D is given by the following formula.
Substituting the above values in equation (i), the minimum distance D can be written as,
Here are three displacements, each measured in meters: ,and. (a) What is ? (b) What is the angle between and the positive z axis? (c) What is the component of along the direction of (d) What is the component of that is perpendicular to the direction of and in the plane of role="math" localid="1658465314757" and (Hint: For (c), consider Eq 3-20. and Fig, 3-18; for (d), consider Eq.3-24.)
A room has dimensions . A fly starting at one corner flies around, ending up at the diagonally opposite corner. (a) What is the magnitude of its displacement? (b) Could the length of its path be less than this magnitude? (c) Greater? (d) Equal? (e) Choose a suitable coordinate system and express the components of the displacement vector in that system in unit-vector notation. (f) If the fly walks, what is the length of the shortest path? (Hint: This can be answered without calculus. The room is like a box. Unfold its walls to flatten them into a plane.)
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