A-Level Mechanics - Vectors
This is Part 2, of the tutorial on vectors, aimed at A-level Mechanics students.
For part 1 click here: http://www.youtube.com/watch?v=LMCHAe4BUHI
These are the 'exam style questions', I will be covering by the end of this tutorial.
- At noon, boat B has the position i+2j km and velocity, 3i+2j km/hr. A lighthouse, L, has the position 5i + j. Find vector LB at 2pm.
- Find LB at 3pm.
- What is the distance between the lighthouse and the boat at 2pm?
- Find LB, at time t hours after noon?
- At what time, will the boat be north of the lighthouse, and what is vector LB at that time?
- Find the distance between the boat and the lighthouse, at t hours after noon.
- At what time will the distance between the boat and the lighthouse be 7km?
Magnitude of a vector[edit | edit source]
Before getting on the hard questions, I'd like to clarify some things first.
Say I have a vector, 3i + 4j. What is the magnitude of this vector?
The magnitude just means the length of the vector (the length of the arrow). Notice you have a right angle triangle here. The base has a length of 3 units, and the height is 4 units. So the hypotenuse would be 5 units (). Therefore the magnitude of the vector 3i +4j, is 5 units. This statement can be mathematically written as: |3i+4j| = 5. (The vertical lines mean magnitude).
Displacement vectors and distances[edit | edit source]
A displacement vector, is essentially an arrow that goes from one co-ordinate to another co-ordinate. Example: Lighthouse A and B have co-ordinates (2,3) and (5,7) respectively. What is the vector AB?
First I shall plot the points:
Answer: Well, AB is a vector going from A to B. From the drawing below you can see that AB = 3i +4j.
Question: What is BA?
If a question asks for the displacement between A and B, it doesn’t specify direction, and so you could write out , or . They would both be correct.
Now, before I continue, I want you to have an appreciation for the fact that a position vector is essentially a co-ordinate. So for example, if you say that object A has co-ordinates (2,3), it’s the same as saying that object A has a position vector (2i+3j). This can be mathematically written as .
Knowing this, you can now take advantage of a simple formula for figuring out displacements vectors.
This formula is:
Also, don’t confuse position vectors with displacement vectors.
Now, what is the distance between points A and B?
So you could do this without knowing anything about vectors.
Just applying Pythagoras' theorem, you can calculate the distance between A and B to be 5 units ().
Or...you could have just figured out the length/magnitude of vectors AB or BA:
Another important thing to know, is that when you are asked for something like “find the position of B relative to boat A”, you are basically being asked for vector AB (NOT vector BA), so the answer would be 3i + 4j. Write down the box below on a separate piece of paper, as it will help with the rest of the questions I’ll be giving you on this page.
- B relative to A = =
Exam Style Questions[edit | edit source]
At noon, boat B has the position i+2j km and velocity, 3i+2j km/hr. A lighthouse, L, has the position 5i + j.
1. Find vector LB at 2pm.[edit | edit source]
Draw it first...
In order to answer the question, you need to figure out where the boat will be at 2pm. For that you use the “position equation”, as I have shown in part 1 of this tutorial.
At 2pm, t=2,
Looking at the diagram, you can see that this is correct...
2. Find LB at 3pm.[edit | edit source]
Thus at 3pm,
Notice how the vector LB is NOT constant. It changes with time.
3. What is the distance between the lighthouse and the boat at 2pm?[edit | edit source]
Simply work out the magnitude of vector LB at 2pm:
4. Find LB, at time t hours after noon?[edit | edit source]
Well, from before we know that at any given time:
This equation tells you what the vector LB is, as a function of time.
Test it out. Question 1 (from before): Find vector LB at 2pm.
At 2pm t=2,
- (Same answer as you got before)
5. At what time, will the boat be north of the lighthouse, and what is vector LB at that time?[edit | edit source]
Draw it out...
You can see that when the boat is north of the lighthouse, vector LB has a zero i component.
From before, ,
We can see that the i component is zero when -4+3t = 0, which rearranges to give t = 4/3. This is equivalent to 1 hour, and 20 minutes. So the answer is 1.20pm.
At that time vector LB would be:
Looking at the diagram above, you can see this about right. Also notice how a similar question was answered in a different way in Part 1 of the tutorial.
6. Find the distance between the boat and the lighthouse, at t hours after noon.[edit | edit source]
Once again, from before, vector LB at any time is LB = (-4+3t)i + (1+2t)j.
To find the distance between L and B, find the magnitude/length of vector LB.
This equation tells you the distance between the boat and the lighthouse as a function of time. Again, you can test it out. Question 3 (from before): What is the distance between the lighthouse and the boat at 2pm?
At 2pm, t=2,
= = 5.39km. (Same answer as you got before)
7. At what time will the distance between the boat and the lighthouse be 7km?[edit | edit source]
Substitute distance = 7,
t = 2.51 hours, and t = - 0.978
Ignore negative time solution.
To convert 2.51 hours to clock time: 2 hours + 60x minutes
= 2 hours 31 minutes.
This is equivalent to 2.31pm.
Final notes[edit | edit source]
- That's it! If you have found this article useful, please comment in the discussion section (at the top of the page). Its just a nice ego boost really...
- Also please comment if there are other questions you want covered, or if there something in this tutorial you did not understand.
- I’d strongly advise you to do all the questions I’ve posted here again by yourself, as it will show you whether you understood it all.
- DRAW!!! It doesn't have to be on square paper. Just a quick sketch (like I've done in part 1 of the tutorial), will do for most questions. When you are able to draw/visualise vectors, things will become a lot easier...
Something extra...[edit | edit source]
Someone has left me a question on how to work out angles between vectors. Here it is...
Question: What is the angle between vector -2i+3j and the unit vector 'i'?
Firstly draw it out:
To find this angle out, you may need to find out a simpler angle first:
Also, make sure not to draw it incorrectly/ work out the wrong angle, like below: