**B + A = A + B**.) Discuss how taking another path to reach the same point might help to overcome an obstacle blocking your other path.

**Exercise 3.16**: Suppose you walk 18.0 m straight west and then 25.0 m straight north. How far are you from your starting point, and what is the compass direction of a line connecting your starting point to your final position? (If you represent the two legs of the walk as vector displacements A and B, as in Figure 3.60, then this problem asks you to find their sum **R = A + B**.)

$30.8 \textrm{ m, }54.2^\circ \textrm{ N of W}$

### Solution video

# OpenStax College Physics, Chapter 3, Problem 17 (Problems & Exercises)

### Calculator Screenshots

*vector B*first and then

*vector A*. So whereas in problem 16 the path first went along part A and then along part B, we're now going to start with

*vector B*and then go along

*A*and we'll confirm that we end up at the same place, so the resultant will be the same. So the

*x*component of the resultant will be the

*x*component of the

*B vector*plus the

*x*component of the

*A vector*. The

*B vector*has no

*x*component since it is vertical and so, the

*A vector*is entirely in the x direction and so, the

*x*component of the resultant is the

*A vector*. That's negative18.0 meters, negative because

*A*is directed left. The

*y*component of the resultant is the

*y*component of the

*B vector*plus the

*y*component of the

*A vector*, the

*A vector*having no

*y*component leaves us with only the

*B vector*which is entirely in the y direction and it's upwards and so it's positive. That's positive 25.0 meters. Now the magnitude of the resultant is the square root of the sum of the squares of its components and so that's the square root of negative 18 meters squared plus 25 meters squared giving 30.8 meters and its direction is the inverse tangent of the size of its

*y*component, divided by the size of its

*x*component. I say size of because I'm referring to these absolute value signs here which means ignore the negative sign if there is one. We'll get our direction by looking at our picture and we'll see that since we're taking the inverse tangent of the

*y*component, divided by the

*x*component, that means we're going to get the angle in here and so our answer will be to the north of west. So we have the inverse tangent of 18.0 meters over 25.0 meters, giving 35.8 degrees north of west. So our resultant is 30.8 meters, 35.8 degrees north of west. What we can see is that if there is an obstacle in the path here, some big rock that you want to avoid, you can get to the same place by taking a different route which is kind of obvious, but now we just sort of demonstrated that that's true using analytical techniques.

## Comments

why did you do 18/25 instead of 25/18

Thanks a lot for pointing this out happyfeet1025. I've flagged this video for a re-take and put a note in the final answer section. I mistakenly substituted the x-component divided by y-component, which results in the angle being to the West of North. The angle is between the resultant vector and the y-axis.

All the best with your studies,

Shaun

This video was updated Nov. 22nd, 2023