Lesson - Vector Addition

Many quantities in the real world do not simply exist as numbers. When your car is moving, it has both speed and direction. To describe the motion of the car, you need some way of writing this information which gives both pieces of information. If you are giving directions in a city, it is not enough to say, "walk four blocks", you have to say walk 2 blocks north and 2 blocks west. Both magnitude and direction are important if you want to get your point across.

A vector is a quantity with both a magnitude and a direction, and it can be written either as a magnitude and a direction (such as 60 miles per hour northwest) or as coordinates (such as 2 blocks west and 2 blocks north).

Adding vectors is a fairly common task. For example, suppose you are flying a plane. Your instruments give you the speed of the plane with respect to the air around you, but the wind is moving with respect to the ground as well! If you want to know how fast you are moving with respect to the ground, you have to determine your relative motion to the ground by adding your velocity with respect to the wind to the wind's velocity with respect to the ground. If the wind is directly at your back, this is easy, but suppose you have a crosswind, how would you do this?

One way of doing this is to draw a picture. You can either draw both vectors starting at the origin, and use them as the sides of a parallelogram, or draw the first vector from the origin, and the second vector starting at the tip of the first. These two graphical techniques are known as the parallelogram and tip-to-tail methods, respectively. The online model that goes along with this lesson will let you use either method.

1. Suppose you are crossing a river at 8 meters per second, and the river is flowing downstream at 6 meters per second. Assuming you are always swimming directly towards the opposite shore, what would the resulting speed and direction with respect to the river bank be?
2. Suppose you are in a plane flying directly north at a speed of 80 knots, and the wind is at your back, but blowing northwest at 20 knots. What is the actual speed and direction your plane flies with respect to the ground?

The drawback to the graphical methods is that they are only as accurate as the person drawing and reading the graph. Also, consider what happens when you have to add not just two vectors, but three, or four, or twenty-thousand. What if you have an equation, and instead of being able to draw the length of a vector, you need to leave it as a variable?

3. Describe a situation where the use of a graphical method would not be the best way to solve a vector addition problem.

When vectors are written in terms of x and y (and z for three dimensions) coordinates, to add two vectors, you simply add up the coordinates. This is easy enough to do for something like city blocks, but what do you do if you do not have clear cut x and y coordinates? You have to find the x and y coordinates for each vector, add components, and if needed solve for the magnitude and direction of the resultant vector.

4. Suppose you have a boat capable of traveling at 20 miles per hour. The river is flowing at 5 miles per hour downstream. If you need to reach a dock directly across from where you are starting, and the magnitude of your boat's velocity with respect to the water is 20 miles per hour, how would you set up an equation to solve for the direction to steer the boat in in the fewest possible steps?
5. What direction would you steer the boat in?
6. How many steps did your solution take?
7. Compare with your classmates, who had the most efficient way of getting the answer, and how did she/he do it?