Vector Addition

Introduction  Vector addition is the process of combining the effects of two or more vectors. For example:

• A helicopter flies 30 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?
• 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?

To solve problems like above, we have to add the vectors. There are a couple ways to do this. One way is to draw a vector diagram. (The other way, using vector components, is purely mathematical, and we will talk about that later.)

Drawing Vector Diagrams

The resultant vector is sometimes called the "net" or "total" vector ("total displacement", "net force", etc.)

Here is a video example showing the whole process of making a vector diagram.

1. Draw a vector diagram for the helicopter problem given in the introduction. Label the resultant with its (real-life) length and direction.

2. Draw a vector diagram for the shark problem given in the introduction. Label the resultant with its (real-life) force and direction.

Click here for answer to Problem 1. (If this doesn't work properly, right-click on the link, and Save the file to your desktop, then play it from there.)

Additional Activities & Practice Draw vector diagrams for each of these situations. Be sure to include a scale and the resultant vector.

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?

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?