Vector Addition Using Components
If you know the components of a vector (either becasue you were told, you drew the vector and measured the components, or you calculated the components using trig), then you can find the components of the resultant by adding the corresponding components of the vectors. In other words...
- the x-component of the resultant is the sum of the x-components of the vectors
- the y-component of the resultant is the sum of the y-components of the vectors
Activities & Practice
For the following problems, consider East to be the positive x-axis, and North to be the positive y-axis. These problems will look familiar. DO NOT draw a scaled diagram. DO draw yourself a quick sketch for each problem to help you visualize the situation, but then calculate the answers purely mathematically. These problems should look familiar.
1. Two sharks are attacking an elephant seal. One shark pulls with a force of 1000 newtons toward the North. The other shark pulls with 700 newtons of force, toward the East. In what direction will the elephant seal be pulled? What is the magnitude of the resultant force? Assume the forces are known to two significant figures.
2. A helicopter
flies 30.0 km North, then 120 km toward azimuth 290°,
then turns again and flies 75 km at azimuth 200°, and then lands.
How far is the landing site from its starting point, and in what direction? SOLUTION VIDEO
3. An airplane is pointed in the direction 35° N of W, flying at 65 m/sec through the air. The air is itself moving relative to the ground, however, at 15 m/sec in the direction 10° E of S. How fast, and in what direction, is the airplane moving relative to the ground?
4. Marvin, a depressed robot, goes off to be by himself and sulk. From his starting point, a picnic table, he travels 45 meters to the west. That spot doesn't make him feel any better, so he turns and travels another 45 meters at 130° azimuth. Still depressed, he turns to 215° azimuth and goes another 15 meters. There he sees a beautiful Monarch butterfly, and feels better.
(a) How far, and in what direction, is Marvin from the picnic table?
(b) If Marvin wanted to return to the picnic table in a single straight line, in what direction would he need to go? SOLUTION
5. Just after leaving Hagrid's shack, Harry Potter is attacked by a dementor. He jumps on his broom and flies 17 km in the direction 33° S of W, with the dementor in hot pursuit. Then he turns and flies 11 km at 20° azimuth. Finally, he turns again and goes another 5 km in the direction 340° mathematical. At this point, Harry finally confronts and defeats the dementor with the help of his Patronus. How far, and in what direction (geographical) is Harry from Hagrid's shack? If Harry wants to return straight to the shack, in what direction (geographical) should he fly? SOLUTION
