Skill 5 of 11

Vector addition

To pull your boat up the stream, you attach a rope and pull from the shore. This will pull the boat towards the shore, though, so you need a friend on the other shore with another rope pulling at another angle. Together these pulls will hopefully overcome the stream’s counter-pull and cause a forwards total force or resulting force:1

$$\vec F_\text{result}= \vec F_\text{you}+ \vec F_\text{friend}+ \vec F_\text{stream}$$

We here simply added vectors together. And adding vectors together simply means adding up their $x$-coordinates and adding up their $y$-coordinates (and their $z$-coordinates when in 3D):

$$\vec F_\text{result}= \vec F_\text{you}+ \vec F_\text{friend}+ \vec F_\text{stream}\quad \Leftrightarrow\\[3ex]
\begin{pmatrix}F_{x,\text{result}} \\ F_{y,\text{result}} \end{pmatrix}= \begin{pmatrix}F_{x,\text{you}} \\ F_{y,\text{you}} \end{pmatrix} + \begin{pmatrix}F_{x,\text{friend}} \\ F_{y,\text{friend}} \end{pmatrix} + \begin{pmatrix}F_{x,\text{stream}} \\ F_{y,\text{stream}} \end{pmatrix} \quad \Leftrightarrow\\[5ex]
\begin{matrix}F_{x,\text{result}} \\ F_{y,\text{result}} \end{matrix} \begin{matrix}= \\ = \end{matrix} \begin{matrix}F_{x,\text{you}} \\ F_{y,\text{you}} \end{matrix} \begin{matrix}+ \\ + \end{matrix} \begin{matrix}F_{x,\text{friend}} \\ F_{y,\text{friend}} \end{matrix} \begin{matrix}+ \\ + \end{matrix} \begin{matrix}F_{x,\text{stream}} \\ F_{y,\text{stream}} \end{matrix}$$

Now there is a total pull of $F_{x,\text{result}}= F_{x,\text{you}} + F_{x,\text{friend}} + F_{x,\text{stream}}$ sideways and a total pull of $F_{y,\text{result}}= F_{y,\text{you}} + F_{y,\text{friend}} + F_{y,\text{stream}}$ along with the stream.

But, hey! Why is the stream force $\vec F_\text{stream}$ added? It counteracts your efforts, so shouldn’t it be subtracted? Hmm, no that wouldn’t make sense: If forces that pull forwards are positive and forces that counteract are negative, then what about your and your friend’s angled forces? Should they be ‘positive-but-also-a-bit-negative’? Signs make sense for single values – scalars – that only have a forwards or backwards sense, which we can call positive and negative, but not for vectors with several more possible directions.

Luckily we don’t have to worry: the coordinates will take care of it! The coordinates depend on the coordinate axes, so we’ll just make sure to include their signs when we input them. The value of $F_{y,\text{stream}}$ will be negative, $F_{y,\text{stream}}<0$, since we chose the $y$-axis forwards.

When thinking of vectors as arrows, we can simply draw the added vectors after one another: where they end, is how far the resulting vector reaches. And it doesn’t matter which you draw first, just like it doesn’t matter if you add ‘4 apples to 5 apples’ or ‘5 apples to 4 apples’.


  1. Sears and Zemansky’s Univesity Physics with Modern Physics’ (book), Hugh D. Young & Roger A. Freedman, Pearson Education, 13th ed., 2012