Tuesday, October 19, 2010

Scalars and Vectors

For students just beginning to study physics, the concept of scalars and vectors is often confusing. We have always just added and subtracted things without problem, but putting fancy names to these concepts can be daunting and makes simple things seem a lot more difficult than they are. Here, I will try to explain what scalars and vectors are.

Of the two, scalars are the simpler quantity. A scalar is a measure of something that has only a magnitude to define it. A 10 meter length of cable is sufficient to define the length, whether we move it 2 meters to the left or 5 meters to the right. Also, 30 seconds is a scalar, and so is 25 degrees Celsius, and 30 liters. The magnitudes of these quantities are all that are needed to define them. Maybe to better understand that, we need to consider vectors as well...

Vectors are quantities that are defined by their magnitude AND their direction. This is the CRITICAL point to differentiate the two. Velocity is defined by the magnitude, but also requires the direction. If you are moving 10 m/s, you are moving 10 m/s IN SOME DIRECTION. This direction is required to fully define your velocity. The same is true when working with forces. You exert a magnitude of force in some direction, which fully defines the force. You can say that vectors are made up of a scalar component and a direction component.

When performing mathematical operations on scalars, you do so as you normally would. Run for 20 seconds, and then run for another 20 seconds, and you have run 40 seconds. Add 1 cup of water, and then add 2 cups, and you have added 3 cups. Since scalars are just composed of magnitude, you can simply operate on them as you always have. In fact, a lot of the math you have ever done, without problems, has been working with scalar quantities. As I said, just adding a fancy name doesn't make it any more difficult.

Working with vectors, however, is slightly more complex. You must always consider the direction component when doing the math. Many introductory physics courses will use examples of airplanes flying to demonstrate vectors. If the plane is flying 200 mph straight north (a vector), and the wind is blowing 50 mph straight east (another vector), you can imagine that the combined effect is that the plane is flying slightly at an angle rather than completely north. Actually, you can determine that the plane is actually travelling 206 mph in a direction of 14 degrees east of straight north, using simple trigonometry that you have already studied! (I will go over vector math in a future post.)

Hopefully, this post has more clearly explained the difference between scalar and vector quantities. Scalars are probably what you are most used to working with normally, whereas vectors require a little more thought, due to their direction component that always must be considered as well. Keep this key difference in mind as you work through your introductory physics homework problems.

Distance vs. Displacement

Continuing on in my posts in Introductory Physics, here I want to explain the difference between DISTANCE and DISPLACEMENT.

First of all, Distance is said to be a "scalar" property, which means that it is a quantity that has no direction associated with it, and is defined only by its magnitude. (Another scalar property is mass... all magnitude, no direction.) Conversely, Displacement is a "vector" property, which means that in addition to its magnitude, it also has a relative direction. (Another vector is velocity... it is defined by a magnitude AND a direction.)

With those definitions, we can differentiate distance and displacement even better. Imagine walking to school in the morning. The number of footsteps you have to take to arrive at school (the magnitude) describes the distance (maybe not the most convenient units, mind you!). If your school is straight up the street, or if you have to take a winding path through the park and over the tracks, the number of steps defines the distance... Direction is irrelevant in explaining the distance. You can walk 100 meters in a straight line, or 1000 meters turning left, right, left, and back again. The measured path you travel is the Distance.

On the other hand, Displacement DOES have direction associated with it. If you are going to school one day, and you go straight there, 100 meters up the street, your displacement is 100 meters North (or whatever direction). Now, on the next day, you have a doctor's appointment first, you have to drive 2000 meters up the street (past your school), and then afterwards you drive back the 1900 meters to your school. The DISPLACEMENT is 100 meters, because you end up 100 meters North of where you started. It doesn't matter the path you take to get there. You could travel around the world and back again, and you would still be displaced 100 meters from where you started. The DISTANCE, however, which has no direction but is just the footsteps (or whatever) you traveled, is much larger. On the first day, the distance is 100 meters. On the second day, you travel 2000 meters plus 1900 meters back, which is 3900 meters.

In summary, think of distance as the path taken to get somewhere, and describe it as some quantity without direction (eg. 50 meters away). Displacement is simply the difference between where you start and where you end, no matter the path taken, but considering the overall direction moved (eg. 25 meters to the right).

Hopefully that helps to explain Distance and Displacement. Drop me a line if you would like any more discussion.

Related Posts