Taking a Hike A hiker begins a trip by first walking 21.0 km southeast from her car. She stops and sets up her tent for the night. On the second day, she walks 46.0 km in a direction 60.0° north of east, at which point she discovers a forest ranger's tower. y (km) Can N W-DE 45.0° 60.0° Tent Tower B x (km) ☹ (a) Determine the components of the hiker's displacement for each day. SOLUTION Conceptualize We conceptualize the problem by drawing a sketch as in the figure. If we denote the displacement vectors on the first and second days by A and B, respectively, and use the ---Select-- as the origin of coordinates, we obtain the vectors shown in the figure. The sketch allows us to estimate the resultant vector as shown. Categorize Drawing the resultant R, we can now categorize this problem as one we've solved before: --Select-- of two vectors. You should now have a hint of the power of categorization in that many new problems are very similar to problems we have already solved if we are careful to conceptualize them. Once we have drawn the displacement vectors and categorized the problem, this problem is no longer about a hiker, a walk, a car, a tent, or a tower. It is a problem about vector addition, one that we have already solved. Analyze Displacement A has a magnitude of Find the components (in km) of A using these equations: AA cos(-45.0°) = Ay A sin(-45.00) = km km km and is directed below the positive x-axis. The ---Select--- ✓ value of Ay indicates the hiker walks in the ---Select---y-direction on the first day. The signs of Ay and Ay also are evident from the figure. Find the components (in km) of B using these equations: BxB cos(60.0°) = By B sin(60.00) = km km (b) Determine the components of the hiker's resultant displacement Ŕ for the trip. Find an expression for R in terms of unit vectors. SOLUTION Use the following equation to find the components (in km) of the resultant displacement R = A + B: Rx = Ax+ Bx = Ry=Ay + By = km km Write the total displacement (in km) in unit-vector form: R= i) km Finalize Looking at the graphical representation in the figure, both components of R are positive, putting the final position in the ---Select---quadrant of the coordinate system. Is this consistent with your answer? EXERCISE