Answer:
distance of 2nd team from 1st team will be: 58.2
Direction of 2nd team from 1st team will be: 14.90 deg North of east
Explanation:
ASSUME Vector is R and makes angle A with +x-axis,
therefore component of vector R is
![R_x = Rcos A](https://tex.z-dn.net/?f=R_x%20%3D%20Rcos%20A)
![R_y = Rsin A](https://tex.z-dn.net/?f=R_y%20%3D%20Rsin%20A)
From above relation
Assuming base camp as the origin, location of 1st team is
away at 21 deg North of west (North of west is in 2nd quadrant, So x is -ve and y is positive)
![R_{1x} = -R_1*cos A_1 = -37*cos 21 deg = -34.54 km](https://tex.z-dn.net/?f=R_%7B1x%7D%20%3D%20-R_1%2Acos%20A_1%20%3D%20-37%2Acos%2021%20deg%20%3D%20-34.54%20km)
![R_{1y} = R_1*sin A_1 = 37*sin 21 deg = 13.25 km](https://tex.z-dn.net/?f=R_%7B1y%7D%20%3D%20R_1%2Asin%20A_1%20%3D%2037%2Asin%2021%20deg%20%3D%2013.25%20km)
location of 2nd team is at
, at 38 deg East of North = 32 km, at 58 deg North of east (North of east is in 1st quadrant, So x and y both are +ve)
![R_{2x} = R_2*cos A_2 = 32*cos 58 deg = 16.95 km](https://tex.z-dn.net/?f=R_%7B2x%7D%20%3D%20R_2%2Acos%20A_2%20%3D%2032%2Acos%2058%20deg%20%3D%2016.95%20km)
![R_{2y} = R_2*sin A_2 = 32*sin 58 deg = 27.13 km](https://tex.z-dn.net/?f=R_%7B2y%7D%20%3D%20R_2%2Asin%20A_2%20%3D%2032%2Asin%2058%20deg%20%3D%2027.13%20km)
Now position of 2nd team with respect to 1st team will be given by:
![R_3 = R_2 - R_1](https://tex.z-dn.net/?f=R_3%20%3D%20R_2%20-%20R_1)
![R_3 = (R_{2x} - R_{1x}) i + (R_{2y} - R_{1y}) j](https://tex.z-dn.net/?f=R_3%20%3D%20%28R_%7B2x%7D%20-%20R_%7B1x%7D%29%20i%20%2B%20%28R_%7B2y%7D%20-%20R_%7B1y%7D%29%20j)
Using above values:
![R_3 = (16.95 - (-34.54)) i + (27.13 - 13.42) j](https://tex.z-dn.net/?f=R_3%20%3D%20%2816.95%20-%20%28-34.54%29%29%20i%20%2B%20%2827.13%20-%2013.42%29%20j)
![R_3 = 51.49 i + 13.71 j](https://tex.z-dn.net/?f=R_3%20%3D%2051.49%20i%20%2B%2013.71%20j)
distance of 2nd team from 1st team will be:
![\left | R_3 \right | = \sqrt (51.49^2 +13.71^2)](https://tex.z-dn.net/?f=%5Cleft%20%7C%20R_3%20%5Cright%20%7C%20%3D%20%5Csqrt%20%2851.49%5E2%20%2B13.71%5E2%29)
![\left | R_3 \right | = 53.28 km = 58.2 km](https://tex.z-dn.net/?f=%5Cleft%20%7C%20R_3%20%5Cright%20%7C%20%3D%2053.28%20km%20%3D%2058.2%20km)
Direction of 2nd team from 1st team will be:
![Direction = tan^{-1} \frac{R_{3y}}{R_{3x}} = tan^{-1}[ \frac{13.71}{51.49}]](https://tex.z-dn.net/?f=Direction%20%3D%20tan%5E%7B-1%7D%20%5Cfrac%7BR_%7B3y%7D%7D%7BR_%7B3x%7D%7D%20%3D%20tan%5E%7B-1%7D%5B%20%5Cfrac%7B13.71%7D%7B51.49%7D%5D)
Direction = 14.90 deg North of east