how do you solve this?

Question

5 paces east (0deg)
7 paces north east (45deg)
10 paces west (180deg)
7 paces north west (135deg)
I graphed it an the resultant is 117deg 11.5 paces
How do you add these vectors with the component method???

Answer 1

We will use Easting (distance, in paces towards the East) and Northing (distance in paces, towards the North).

Given a magnitude (total number of paces in one vector) and a direction Z (in degrees, in this case), we use the formulas:

E = Magnitude * Cos(Z)
N = Magnitude * SIn(Z)
Do keep track of the signs. Going west gives a negative easting.
And I did not make up these words (Easting and Northing)

5 paces east (Z = 0)
E = +5
N = 0

7 paces NE (Z=45)
E = 7*Cos(45) = +5 (I've rounded)
N = 7*Sin(45) = + 5

10 paces west (Z=180)
E = 10*Cos(180) = -10
N = 10*Sin(180) = 0

and so on.

Then add up all the Eastings and all the Northings.

The magnitude of the resultant vector will be the "distance"
SQRT( E^2 + N^2).

The direction will be given by the arctangent

Tan(Z) = N/E

(You may have to resolve the ambiguity by figuring out the quadrant -- easy to do by checking the signs of E and N)