Vector Addition Problem: What is the Fourth Vector?

[armolinasf]

Homework Statement

a person goes 180 m due west then 210 m 45 degrees east of south then 280 m 30 degrees east of north. After a fourth displacement the person is back where they started. what is the fourth vector?

The Attempt at a Solution

So I first added the first two vectors of 180 and 210 using the cosine rule giving me 223 meters then added that sum with the third to get the fourth again using the cosine rule this gave me 272 m. However, the answer in my book says it should be 144m, what did I do wrong? Thanks for the help!

 

[lanedance]

show how you added them, and i'll look at your work

i think writing out in terms of components will be easiest, for example if N & E are positive:
the first vector 180W = (0,180)
the 2nd will be 210SE = 210*(-cos(45),sin(45))

 

[armolinasf]

my first vector was sqrt(180^2+210^2-189_210*cos45)=223 then using this vector I was able to calculate the fourth one by sqrt(223^2+280^2-223*280cos30)=272

 

[Born2bwire]

I don't understand your process. Those aren't vectors, a vector has a direction and a magnitude. Look at how lanedance defined the vectors for the displacement above. Write out the three vectors that describe each of the three displacements. Then add the vectors to find the total displacement. Using this you can find the fourth vector that takes you back to the starting point.

So what are the three vectors and the vector for the total displacement?

 

[armolinasf]

I'm trying to do it without components

 

[lanedance]

do you have to? it'll be a lot more difficult without using components directly...

 

[armolinasf]

I don't have to but id like to be able to know that I can do it either way.

 

[lanedance]

i would stick with components, then convert to length and direction at the end.

even when you think you are adding vectors without components in effect you are doing the same thing just with a different basis and converting back to length and direction at the end

 

[armolinasf]

alright I'll add it up with components to see if i get the same answer as the book