Chapter 1: Problems (1.67 - 1.71)

1.67 You are to program a robotic arm on an assembly line to move in the xy-plane. Its first displacement is $\vec{A}$; its second displacement is $\vec{B}$, of magnitude 6.40 cm and direction $63^\circ$ measured in the sense from the +x-axis toward the -y-axis. The resultant $\vec{C} = \vec{A} + \vec{B}$ of the two displacements should also have a magnitude of 6.40 cm but a direction $22.0^\circ$ measured in the sense from the +x-axis toward the +y-axis. a) Draw the vector addition diagram for these vectors, roughly to scale. b) Find the components of $\vec{A}$. c) Find the magnitude of $\vec{A}$.

1.68 a) Find the magnitude and direction of the vector $\vec{R}$ that is the sum of the three vectors $\vec{A}, \vec{B},$ and $\vec{C}$ in Fig. 1.28. In a diagram, show how $\vec{R}$ is formed from the three vectors in Fig. 1.28. b) Find the magnitude and direction of the vector $\vec{S} = \vec{C} - \vec{A} - \vec{B}$. In a diagram, show how $\vec{S}$ is formed from the three vectors in Fig. 1.28.

1.69 As noted in Exercise 1.33, a spelunker is surveying a cace. She follows a passage 180 m straight west, then 210 m in a direction $45^\circ$ east of south, then 280 m at $36^\circ$ east of north. After a fourth unmeasured displacement, she finds herself back where she started. Use the method of components to determine the magnitude and direction of the fourth displacement. Draw the vector addition diagram and show that it is in qualitative agreement with your numerical solution.

1.70 A sailor in a small sailboat encounters shifting winds. She sails 2.00 km east, then 3.50 km southeast, and then an additional distance in an unknown direction. Her final position is 5.80 km directly east of the starting point (Fig. 1.32). Find the magnitude and direction of the third leg of the journey. Draw the vector addition diagram and show that it is in qualitative agreement with your numerical solution.

1.71 A cross-country skier skis 2.80 km in the direction $45.0^\circ$ west of south, then 7.40 km in the direction $22.0^\circ$ south of west. a) Show these displacements in a diagram. b) How far is the skier from the starting point?