Question

On a safari, a team of naturalists sets out toward a research station located 4.63 km away in a direction 38.7 ° north of east. After traveling in a straight line for 2.13 km, they stop and discover that they have been traveling 25.9 ° north of east, because their guide misread his compass.
What are (a) the magnitude and (b) the direction (as a positive angle relative to due east) of the displacement vector now required to bring the team to the research station?

Answer 1

Exactly, it is a vector subtraction problem.  Let t=theta,

t1=38.7, d1=4.63;  t2=25.9, d2=2.13

v1=d1 <cos(t1), sin(t1)>=<3,6134, 2.8949>

v2=d2 <cos(t2), sin(t2)>=<1.9161, 0.9304>

Final vector

v3 = v1-v2

=<v3x, v3y>

=<1.6973, 1.9645>

where

v3x=d1*cos(t1)-d2*cos(t2)

v3y=d1*sin(t1)-d2*sin(t2)

The final vector v3 has therefore a magnitude of

||v3||=sqrt(1.6973^2+1.9645^2)=2.5962, and a direction of

theta=atan(1.9645/1.6973)=49.17 degrees  north of east

Note: all three vectors are in the direction of the first quadrant.