Class content I > The Main Question: Motion > Kinematics

3.1.2.1.1

Prerequisite:

• Adding vectors

Note: In our discussion below we have sometimes written vectors with arrows over them (or hats if they are unit vectors) and sometimes written them as bold. These two notations are common and depend on how the display is being handled. When we do pictures, we can use the arrows. When we are using web display (HTML) we use bold. This can be confusing, but you'll just have to get used to it.

To see how the addition of vectors works in a particular case, let's consider the following problem:

Given that

For the following vector operation, find the results both algebraically and show the meaning geometrically.

Let's solve this algebraically first. To do it, we just replace the vectors by how they are expressed in terms of the unit vectors and then rearrange to collect terms.

Note two tricky bits that are obvious once you see them but can be confusing when you are getting started:

1. When we have just a unit vector by itself (i or j) we mean "1" times that vector. If we're adding we have to put it in.
2. When we have no component in a vector, it corresponds to 0 amount of that vector.

These are perhaps more obvious when we use the parenthesis notation that is commonly used when you are doing geometry in an algebra class: (xi + yj) → (x,y). (This is only convenient if we are never going to change what we mean by i or j. Since in this class we do want to be free to use different choices of directions as our unit vectors, we will mostly use i and j.)

The same answer as we got before, just expressed in the other notation.

Now let's see what it looks like geometrically.

Conceptually, the idea is to use to displacement metaphor: the idea of vectors is built on the physical idea of displacement. You start from the tail and move to the head. Adding vectors is like doing the displacements one after the other. So that you put the tail of the second at the head of the first and follow it to the second head. The sum is the displacement from the first tail to the second head.

In our case we'll do it in two steps since we have three vectors to add. The geometry looks like this:

In the left panel above we've added 2a + b. We've drawn the original vectors a and b in lightly in blue and red respectively. We've then doubled a (deep blue) and shifted b so that its tail starts at the head of 2a -- just as if you had walked from the origin to the head of the vector 2a and then started to walk the vector b. The result, shown in green, is 2a+b.

Next we add the vector c to that result. Again, we draw the original vector in lightly (in purple) and shift it so it follows on as a tail-to-head displacement beginning at the end of 2a+b. The result, shown in black, is the same as got from our algebraic analysis.

Joe Redish 12/26/14