Finding Resultant Vector

Question

For some reason when I try to calculate the resultant vector's magnitude using the "break into X and Y components" method, I keep getting the wrong answer. In the textbook, the answer is said to be $13$ (which implies that the triangle which is formed is right-angled), however, I just keep getting $\sqrt{229} = 15.13$ as my answer by using this method. I can't spot the problem that I keep making.

The question (from the text book):

Here's what I have done:

• The small triangles that I formed are based on the given forces in the question.

Thanks a lot!

Answer 1

$$\vec{r}_x = \color{red}- 12 \cos 75^\circ + 5 \sin 15^\circ$$ because the 12 vector is going to the left, not the right.

Answer 2

Your mistake stems from the fact that when calculating $\vec{r}_x$, you add the two components. However, if you take a look at the two triangles you've drawn, you'll notice that they're facing in opposing directions (that is, one of the components of $\vec{r}_x$ goes to the right and the other goes to the left). As such, one of them should be negative (relative to the other).