Vectors \(\overrightarrow{\mathbf{A}}\) and \(\overrightarrow{\mathbf{B}}\) have equal magnitudes of 5.00. The sum of \(\overrightarrow{\mathbf{A}}\) and \(\overrightarrow{\mathbf{B}}\) is the vector \(6.00 \hat{\mathbf{j}}\). Determine the angle between \(\overrightarrow{\mathbf{A}}\) and \(\overrightarrow{\mathbf{B}}\).

You are spending the summer as an assistant learning how to navigate on a large ship carrying freight across Lake Erie. One day, you and your ship are to travel across the lake a distance of 200 km traveling due north from your origin port to your destination port. Just as you leave your origin port, the navigation electronics go down. The captain continues sailing, claiming he can depend on his years of experience on the water as a guide. The engineers work on the navigation system while the ship continues to sail, and winds and waves push it off course. Eventually, enough of the navigation system comes back up to tell you your location. The system tells you that your current position is 50.0 km north of the origin port and 25.0 km east of the port. The captain is a little embarrassed that his ship is so far off course and barks an order to you to tell him immediately what heading he should set from your current position to the destination port. Give him an appropriate heading angle.

A person going for a walk follows the path shown in Figure P3.35. The total trip consists of four straight-line paths. At the end of the walk, what is the person’s resultant displacement measured from the starting point?

A ferry transports tourists between three islands. It sails from the first island to the second island, 4.76 km away, in a direction \(37.0^{\circ}\) north of east. It then sails from the second island to the third island in a direction \(69.0^{\circ}\) west of north. Finally it returns to the first island, sailing in a direction \(28.0^{\circ}\) east of south. Calculate the distance between (a) the second and third islands and (b) the first and third islands.

Two vectors \(\overrightarrow{\mathbf{A}}\) and \(\overrightarrow{\mathbf{B}}\) have precisely equal magnitudes. For the magnitude of \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}\) to be 100 times larger than the magnitude of \(\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}\0, what must be the angle between them?

Two vectors \(\overrightarrow{\mathbf{A}}\) and \(\overrightarrow{\mathbf{B}}\) have precisely equal magnitudes. For the magnitude of \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}\) to be larger than the magnitude of \(\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}\) by the factor n, what must be the angle between them?

Review. The biggest stuffed animal in the world is a snake 420 m long, constructed by Norwegian children. Suppose the snake is laid out in a park as shown in Figure P3.39, forming two straight sides of a \(105^{\circ}\) angle, with one side 240 m long. Olaf and Inge run a race they invent. Inge runs directly from the tail of the snake to its head, and Olaf starts from the same place at the same moment but runs along the snake. (a) If both children run steadily at 12.0 km/h, Inge reaches the head of the snake how much earlier than Olaf? (b) If Inge runs the race again at a constant speed of 12.0 km/h, at what constant speed must Olaf run to reach the end of the snake at the same time as Inge?

Ecotourists use their global positioning system indicator to determine their location inside a botanical garden as latitude 0.002 43 degree south of the equator, longitude 75.642 38 degrees west. They wish to visit a tree at latitude 0.001 62 degree north, longitude 75.644 26 degrees west. (a) Determine the straight-line distance and the direction in which they can walk to reach the tree as follows. First model the Earth as a sphere of radius \(6.37 \times 10^6\) m to determine the westward and northward displacement components required, in meters. Then model the Earth as a flat surface to complete the calculation. (b) Explain why it is possible to use these two geometrical models together to solve the problem.

A vector is given by \(\overrightarrow{\mathbf{R}}\) = \(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}}\). Find (a) the magnitudes of the x, y, and z components; (b) the magnitude of \(\overrightarrow{\mathbf{R}}\); and (c) the angles between \(\overrightarrow{\mathbf{R}}\) and the x, y, and z axes.

You are working as an assistant to an air-traffic controller at the local airport, from which small airplanes take off and land. Your job is to make sure that airplanes are not closer to each other than a minimum safe separation distance of 2.00 km. You observe two small aircraft on your radar screen, out over the ocean surface. The first is at altitude 800 m above the surface, horizontal distance 19.2 km, and \(25.0^{\circ}\) south of west. The second aircraft is at altitude 1100 m, horizontal distance 17.6 km, and \(20.0^{\circ}\) south of west. Your supervisor is concerned that the two aircraft are too close together and asks for a separation distance for the two airplanes. (Place the x axis west, the y axis south, and the z axis vertical.)

Review. The instantaneous position of an object is specified by its position vector leading from a fixed origin to the location of the object, modeled as a particle. Suppose for a certain object the position vector is a function of time given by \(\overrightarrow{\mathbf{r}}\) = \(4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-2 t \hat{\mathbf{k}}\), where \(\overrightarrow{\mathbf{r}}\) is in meters and t is in seconds. (a) Evaluate \(d \overrightarrow{\mathbf{r}} / d t\). (b) What physical quantity does \(d \overrightarrow{\mathbf{r}} / d t\) represent about the object?

Vectors \(\overrightarrow{\mathbf{A}}\) and \(\overrightarrow{\mathbf{B}}\) have equal magnitudes of 5.00. The sum of \(\overrightarrow{\mathbf{A}}\) and \(\overrightarrow{\mathbf{B}}\) is the vector \(6.00 \hat{\mathbf{j}}\). Determine the angle between \(\overrightarrow{\mathbf{A}}\) and \(\overrightarrow{\mathbf{B}}\).

A rectangular parallelepiped has dimensions a, b, and c as shown in Figure P3.45. (a) Obtain a vector expression for the face diagonal vector \(\overrightarrow{\mathbf{R}}_1\). (b) What is the magnitude of this vector? (c) Notice that \(\overrightarrow{\mathbf{R}}_1\), \(c \hat{\mathbf{k}}\), and \(\overrightarrow{\mathbf{R}}_2\) make a right triangle. Obtain a vector expression for the body diagonal vector \(\overrightarrow{\mathbf{R}}_2\).

A pirate has buried his treasure on an island with five trees located at the points (30.0 m, -20.0 m), (60.0 m, 80.0 m), (-10.0 m, -10.0 m), (40.0 m, -30.0 m), and (-70.0 m, 60.0 m), all measured relative to some origin, as shown in Figure P3.46. His ship’s log instructs you to start at tree A and move toward tree B, but to cover only one-half the distance between A and B. Then move toward tree C, covering one-third the distance between your current location and C. Next move toward tree D, covering one-fourth the distance between where you are and D. Finally move toward tree E, covering one-fifth the distance between you and E, stop, and dig. (a) Assume you have correctly determined the order in which the pirate labeled the trees as A, B, C, D, and E as shown in the figure. What are the coordinates of the point where his treasure is buried? (b) What If? What if you do not really know the way the pirate labeled the trees? What would happen to the answer if you rearranged the order of the trees, for instance, to B (30 m, -20 m), A (60 m, 80 m), E (-10 m, -10 m), C (40 m, -30 m), and D (-70 m, 60 m)? State reasoning to show that the answer does not depend on the order in which the trees are labeled.