Just about anything you measure in Physics can be divided into two categories: scalars and vectors.
Scalars: Any measurement that is given as a single number, and nothing else. It has magnitude, but no direction.
Vectors: A measurement that is given as a number and a direction. It has magnitude and direction.
- We often use arrows to represent vectors. In fact, for the rest of the course you should see them as being interchangeable.
- If I wanted to show you a vector for a car moving at 20km/h [East], and another one moving at 10km/h [East], it might look something like this…
- Notice that the 10km/h vector is half the size of the 20km/h vector.
- The size of the vector is supposed to accurately represent the magnitude of whatever it is showing. That way, any diagrams you draw will be correctly to scale.
- If you want, you can even draw all your vector diagrams to scale and solve them by measuring stuff... although calculations will usually give you more accurate answers.
Let’s say you wanted to draw a single vector showing the displacement of an object as it moved from its initial position (d1 )to a final position (d2).
- Draw a straight arrow between the two points; this is the displacement.
- It is a vector because it has a magnitude and a direction.
- I have used the symbol Δd to represent the displacement, since it is a change in the position of the object.
What if there were a bunch of vectors going from d1 to d2?
- It still results in the same thing… moving from d1 to d2.
- We don’t pay attention to all the extra moving around that was done. We are only concerned with the initial and final positions.
- The one vector that represents all the other vectors is called a resultant vector.
- This is the same as the one vector that we drew in the example above.
- Any time you are adding vectors together, you are trying to find this resultant, a vector that shows what all of the others would result in if put together. We’ll look at this in more detail later.
Vector Directions
We need a system that we can use to give the direction these vectors are pointing in.
- We can't really use a math system (since they usually assign any old line as a reference point), or a navigation method (since they assign a reference point for no good reason).
- Instead, we'll rely on a system of reference points everyone can agree on... North, East, West, and South.
- In this system we will use one of the four compass points as our reference, and then measure how many degrees towards one of the other compass points we have moved.
Example 1: What is the direction of the vector shown here?
If you look at where the angle is placed in this diagram, you'll probably agree that we are measuring an angle of 30° away from the North.
In fact, the 30° angle is moving towards where East is.
In the physics style of giving a direction, we would write [N30°E], which is read as "North 30° East". The reference line (North) is given first, and then the number of degree away from it (30°) going towards another of the reference lines (East).
We always put directions in [square brackets] to show that they are the direction, and not some weird formula!
Since all the way from North to East is a full 90°, we know that this could also be drawn showing that the vector is 60° (we get it from 90° - 30°) counterclockwise from the East. This would mean that we could also measure this vector as [E60°N] . Of the two [N30°E] is considered more, ummm, polite, since the angle is smaller. Either one is still correct.
There is an older version of this system that still pops up from time to time. In it the order is changed to (1) the number of degrees, (2) towards a reference, (3) from a reference. The example above would be written as [30°E of N] which is read as "30°East of North". We will avoid using this older system.
Try each of the following examples. By clicking on the check mark below and to the right of each you will be able to see if you are correct.