4 Vector Addition
Last chapter we saw vectors are best thought of as instructions, e.g. “move 1 down and 2 to the right.” What makes vectors useful is that we can combine vectors. Let’s see how this works:
Exercise 4.1: Vector Addition
You are given two displacement vectors 
and 
.
If you had to guess how to add 
and 
, what would you write down?
To figure out what this means physically, consider the displacement vector
= “follow and then follow .”
Reason what the quantities 
and 
that describe the vector should be in terms of the components of and . Justify your answer in English.
There are several things to highlight from the above discussion. First,
To add vectors, we just need to add the components.
In other words, 
. This is pretty much the only thing you can write that makes sense, so it’s easy to remember!
The second thing to take away is that
When adding vectors, the “+” sign means “and then.”
That is, in English, the equation 
means “Follow vector and then follow vector .” This brings me to the last important thing I want to highlight:
In physics, every equation can be thought of as a sentence in English!
A very big part of physics is being able to “translate” English into math and vice-versa: math is a tool, but understanding happens in English.
This understanding allows us to figure out how to graphically add vectors as well, as we now show.
Exercise 4.2: Adding Vectors Graphically.
Consider the diagram to the right. Find the vector where 
.
Do not use components! Try solving this problem by drawing a picture instead. Remember, the vector is found by following , and then following .
If you’re not sure how to get started, take a look at the solution video below. You’ll get a few more chances to practice after watching the video.
Solution:
Exercise 4.3: Practice Vector Addition
Exercise 4.4: Practice Vector Addition
Exercise 4.5: Equations are Sentences!
I mentioned that in physics all equations should be thought of as sentences in English. Let’s drive this point home.
In the diagram below, you are standing at the origin (brown circle). Your two friends Charlie (C) and Beth (B) are nearby.
Key Takeaways