Final answer:
Graphically, Mr. Smith's displacement is determined by measuring the diagonal of a north and east directed vector plot. For the sailboat, vector math combines the westward and southward velocities to find magnitude and direction of the resultant.
Step-by-step explanation:
To solve the hiking problem graphically, you would use a ruler and a protractor. Following the given directions, draw a vector 110 km north and then from the end of this vector, a second vector 80 km east. To find how far Mr. Smith is from his starting point, measure the diagonal distance from the start point to the end of the second vector—this is the magnitude of the resultant vector. For the direction, if we consider north as the vertical axis and east as the horizontal axis, we would measure the angle between the eastward axis and the resultant vector using a protractor turned upside down, since the vector is south of the eastward pointing axis.
For the sailboat problem, we can solve using vector math. The resultant velocity vector ·V can be calculated by combining the two given vectors: 45 km/hr to the west and 18 km/hr to the south. Using Pythagorean theorem, the magnitude of ·V is √((45)^2 + (18)^2) = √(2025 + 324) = √2349 km/hr. The direction can be found using trigonometry, specifically the tangent function, to find the angle θ that the resultant vector makes with the western axis (which can be considered the negative x-axis). The tangent of the angle is the opposite side over the adjacent side, so tan(θ) = 18/45. From here you can find θ using the inverse tangent function, which will give the direction.