Vectors and Magnitude

Question

Steve is driving in his car to take care of some errands. The first errand has him driving to a location 3 km West and 6 km North of his starting location. Once he completes that errand, he drives to the second one which is 7 km East and 6 km South of the first errand. What is the magnitude of the vector that describes how far the car has traveled from its starting point?

Answer 1

Remember, the directions you are given are also vectors. You can think of these directions as an x-y plane with north as positive y, west as negative x, etc.

There are really two ways to do this (the second is probably the best but I would recommend understanding the first one too).

1) Visually: The first one in the problem is a vector pointing due-West with a magnitude of 3 km. When we add vectors, which is what you are asked to do here, the visual way is to draw the two vectors. So first draw a vector with an arrow pointing west and label it as 3km. Draw the next vector, starting at the first one's arrow, pointing north and label that as 6km.

You can then find what their sum is by drawing another line from the starting place of the first to the end arrow of the second. You'll notice you have a right-triangle. You can use trig and the magnitudes of the west and north vectors to figure out this new vector's magnitude and angle (best to compare it with the directions given, so in this case, "some degree" north of west).

Use the same technique to figure out the second vector sum with the 7km east and 6km south. To answer the final question, you'll have to add these two summed vectors together. That summation, particularly with more difficult problems, can get a bit complicated as you will likely not have a nice right triangle.

2) Algebraically: You can think of these four vectors as all overlapping on top of one another with their respective magnitudes and directions. This works particularly well when you have perpendicular directions like those in the problem.

Try to simplify your thinking to just one dimension at a time. If you drove 6km North and then went 6km South, where would you be? Again, you can equate north to positive y and south to negative y. So this is just the same as asking what 6 minus 6 equals. Thus you know, very easily, that in terms of north-south movement, Steve is right back where he began. Try the same thing with the East and West vectors!

Hope that helps! Let me know if you have any other questions.