Chapter 1

You are lost at night in a large, open field. Your GPS tells you that you are 122.0 m from your truck, in a direction 58.0${}^{\circ }$ east of south. You walk 72.0 m due west along a ditch. How much farther, and in what direction, must you walk to reach your truck?

A ship leaves the island of Guam and sails 285 km at 62.0${}^{\circ }$ north of west. In which direction must it now head and how far must it sail so that its resultant displacement will be 115 km directly east of Guam?

A physical therapy patient has a forearm that weighs 20.5 N and lifts a 112.0-N weight. These two forces are directed vertically downward. The only other significant forces on this forearm come from the biceps muscle (which acts perpendicular to the forearm) and the force at the elbow. If the biceps produces a pull of 232 N when the forearm is raised 43.0${}^{\circ }$ above the horizontal, find the magnitude and direction of the force that the elbow exerts on the forearm. (The sum of the elbow force and the biceps force must balance the weight of the arm and the weight it is carrying, so their vector sum must be132.5 N, upward.)

You decide to go to your favorite neighborhood restaurant. You leave your apartment, take the elevator 10 flights down (each flight is 3.0 m), and then walk 15 m south to the apartment exit. You then proceed 0.200 km east, turn north, and walk 0.100 km to the entrance of the restaurant. (a) Determine the displacement from your apartment to the restaurant. Use unit vector notation for your answer, clearly indicating your choice of coordinates. (b) How far did you travel along the path you took from your apartment to the restaurant, and what is the magnitude of the displacement you calculated in part (a)?

While following a treasure map, you start at an old oak tree. You first walk 825 m directly south, then turn and walk 1.25 km at 30.0${}^{\circ }$ west of north, and finally walk 1.00 km at 32.0${}^{\circ }$ north of east, where you find the treasure: a biography of Isaac Newton! (a) To return to the old oak tree, in what direction should you head and how far will you walk? Use components to solve this problem. (b) To see whether your calculation in part (a) is reasonable, compare it with a graphical solution drawn roughly to scale.

A fence post is 52.0 m from where you are standing, in a direction 37.0${}^{\circ }$ north of east. A second fence post is due south from you. How far are you from the second post if the distance between the two posts is 68.0 m?

A dog in an open field runs 12.0 m east and then 28.0 m in a direction 50.0${}^{\circ }$ west of north. In what direction and how far must the dog then run to end up 10.0 m south of her original starting point?

Ricardo and Jane are standing under a tree in the middle of a pasture. An argument ensues, and they walk away in different directions. Ricardo walks 26.0 m in a direction 60.0${}^{\circ }$ west of north. Jane walks 16.0 m in a direction 30.0${}^{\circ }$ south of west. They then stop and turn to face each other. (a) What is the distance between them? (b) In what direction should Ricardo walk to go directly toward Jane?

You are camping with Joe and Karl. Since all three of you like your privacy, you don't pitch your tents close together. Joe's tent is 21.0 m from yours, in the direction 23.0${}^{\circ }$ south of east. Karl's tent is 32.0 m from yours, in the direction 37.0${}^{\circ }$ north of east. What is the distance between Karl's tent and Joe's tent?

In the methane molecule, CH${}_4$, each hydrogen atom is at a corner of a regular tetrahedron with the carbon atom at the center. In coordinates for which one of the C$-$H bonds is in the direction of $\hat{ı}$ + $\hat{ȷ}$ + $\hat{k}$, an adjacent C$-$H bond is in the $\hat{ı}$ $-$ $\hat{ȷ}$ $-$ $\hat{k}$ direction. Calculate the angle between these two bonds.