5 Using Vectors
We have learned what vectors are and how to add them. We are now in a position to start solving problems using vectors. Let’s try it.
Exercise 5.1: Using Vectors to Describe Displacements
Adam is camping. He leaves his camp for an afternoon walk. He walks 1 mile eastward, turns left at a 45 degree angle, and walks for another 2 miles. From there, Adam heads straight back to camp.
How long was the last leg of Adam’s walk, and from what direction did he arrive onto his campsite?
We will go through a step-by-step guide of how to solve this problem. The method used here can be used to solve every problem in this book.
Step 1- Draw a picture: Make a digram that clearly shows what’s happening in this story. You will need to go sentence by sentence, and make sure your diagram includes all of the information given in each sentence. Make sure everything is properly labelled.
Try doing the best picture your can, and then look at my solution video. Note “best” doesn’t mean “most beautiful.” A good picture in physics is one that clearly tells the full story of the problem.
Step 2- Find Relations: With your picture in hand, go through the story of the problem to find relations between the quantities that appear in your drawing. In this case, the story is simple:
Adam goes along 
, then along 
, and finally along 
, at which point he is back where he started.
You should be able to turn this little story into an equation that relates the vectors , , and . Once you have your vector equation, remember:
A vector equation is really one equation for each of the components.
That is, the relation you derived must be true for both the 
-components of the vectors, and the 
-components of the vectors. Use this to turn your vector equation into two ordinary equations. Stick to variable names here: do not use numbers.
If you get stuck, go through my solution below for step 2.
Step 3- Solve Unknowns: At the end of step 2 you derived relations between the variables of the problem. Some of these variables are known (e.g. 
, 
), and some unknown (e.g. 
and 
). At this point, you can usually just solve for the unknown variables, so you might as well do so.
With the unknowns in hand, answering the question posed in the problem is always trivial.
Important: Do all your algebra using variables, not numbers.
Try it now. If you get stuck, you can view my solution before.
I will refer to the above problem-solving framework as the three steps of problem solving:
1. Draw a picture.
2. Find relations.
3. Solve unknowns.
Every problem in this book can be solved using this framework!
Later in the book I will add a “step 0,” though this step 0 is really more of an addendum to step 1.
One thing to notice is that the question asked is not relevant to how we find the solution. This is basically always true: once you have the unknown quantities in the problem, you can answer any question you are asked!
Let’s do two more problems to make sure we got it.
Exercise 5.2: Solving Vector Problems
A few friends are sailing on a lake. The sailboat first travels 800 m south. It then travels 400 m at an angle 30° north of east. Finally, the sailboat travels 250 m at an angle 60° east of south. How far from its starting point is the sailboat, and in which direction?
A. Draw a picture, going sentence by sentence. Remember to give names and label everything in your drawing that is relevant to the story of the problem.
You can see the solution below.
B. Find relations. Use the story of the problem to find how the vectors you drew are related. Then, turn the vector equation into equations for the vector components. Remember: use names, not numbers. (Solution below).
C. Solve unknowns. Remember to do the algebra in terms of variables. Numbers are only to be plugged in at the very end.
Exercise 5.3: Solving Vector Problems
Joe leaves home heading west. He walks 1 mi to a nearby pub and gets hammered. Joe and his friends leave the pub for a different pub, but Joe is too plastered to know how far or in which direction he walked. At the end of the night, one of his friends points Joe towards his house, which is 2.5 mi North-East of this second pub. What is the distance between the first and second pub that Joe visited that day?
If you get stuck, use the slider to reveal the picture I drew for this problem. If your picture doesn’t match, fix it, and then try finishing it up.
If you’re still stuck, try using this hint.
You may be wondering why I didn’t give you the solution to this last problem. The reason is that it is easy to fool yourself into thinking you understand the solution when you see the solution. However- if you can’t replicate it on your own, that is a clear signal that you need to spend more time solving problems. If you had to use the hints to solve the last problem, I recommend you try some of the optional practice problems below.
Key Takeaways
Practice Problems:
PP 5.1: An airplane needs to make two short flights. The first leg is 60 mi long, heading 
North of East. For the second leg, the plane travels 80 mi at an angle of 30 degrees East of South. Find: A) how far the plane is from its starting point; and B) the angle that the plane’s final position vector makes with the -axis.
PP 5.2: An ant goes foraging for food. It first walks 2 m at an angle of 15 degrees South of West. It stops, and it starts heading East for 3 m. The ant stops once more, turn to a heading of 30 degrees West of North, and walks for an additional 1 m. At this point the ant finds some food, grabs it, and heads straight back to its nest. How far did the ant have to travel to its nest after finding the food?