Class content I > The Main Question: Motion > Kinematics

3.1.2.1.2

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.

Prerequisite:

• Adding vectors
• Example: Vector addition

Subtraction of vectors works just like addition, but to subtract a vector, b, you add the vector that looks just like b  but with all the signs of its components reversed. Here's an example

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.

To subtract b all we had to do was change the -3 to a +3 since a minus times a minus is a plus. The rest is the same as adding.

This also looks simpler in the parenthesis notation that is commonly used when you are doing geometry in an algebra class: (xi + yj) → (x,y). 

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 a displacement metaphor but now we have to add -b instead of just b. The geometry looks like shown at the right. We've drawn the original vectors a and b in lightly in blue and red respectively. We've then doubled a (deep blue) and flipped b so that it becomes -b and 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, the same as we got from our algebraic analysis.

Joe Redish 12/26/14