\(\def \u#1{\,\mathrm{#1}}\) \(\def \abs#1{\left|#1\right|}\) \(\def \ast{*}\) \(\def \deg{^{\circ}}\) \(\def \tau{\uptau}\) \(\def \ten#1{\times 10^{#1}}\) \(\def \redcancel#1{{\color{red}\cancel{#1}}}\) \(\def \BLUE#1{{\color{blue} #1}}\) \(\def \RED#1{{\color{red} #1}}\) \(\def \PURPLE#1{{\color{purple} #1}}\) \(\def \th#1,#2{#1,\!#2}\) \(\def \lshift#1#2{\underset{\Leftarrow\atop{#2}}#1}}\) \(\def \rshift#1#2{\underset{\Rightarrow\atop{#2}}#1}}\) \(\def \dotspot{{\color{lightgray}{\circ}}}\) \(\def \ccw{\circlearrowleft}\) \(\def \cw{\circlearrowright}\)
Appendix B: Vectors
2.

Adding Vectors

Like with numbers, there are certain operations we can perform on vectors. One important example is that we can add vectors together. To add two vectors together, we move them so that they form a chain; the sum of the vectors is the vector from the beginning of the chain to the end of the chain. It is important, when using this method, that the lengths of the arrows be proportional to their magnitudes, or it won't work right.

It doesn't matter what order you put the arrows in when you build the chain; the result is the same.

You can extend this to any number of vectors, building a longer chain.

To subtract one vector from another, we flip the second vector before building the chain.

We occasionally offset the vectors a little bit, like in this second example, when they would otherwise overlap each other.

If the vectors build a chain that form a closed loop, then we say that they add up to the zero vector, written \(\vec 0\). For instance, \(\vec A-\vec A=\vec A+(-\vec A)=\vec 0\), as we might expect.