Step-by-step Answer:
For both problems, we need the results of vector arithmetic.
Q13
The resultant motion is the sum of the two vector, boat and current.
Sum of the x-components:
Vx=20cos(45)-6cos(60)=11.142
Vy=20sin(45)-6sin(60)=8.946
Magnitude = sqrt(Vx^2+Vy^2)=14.29 knots
Angle arctan(Vy/Vx)=arctan(8.946/11.142)=arctan(0.803)=38.76 degrees (from the x-axis) i.e. 39 degrees north of east.
Expressed in East of north, it is 90-39=51 degrees east of north.
Q14
V=velocity of airplane (vector)=570 mph due east=<-570,0>
W=velocity of wind (vector)
V+W=velocity with wind = 581 mph 87 degrees W of N
= < -581sin(87), 581cos(87)>
Velocity of wind = (V+W) - V
=<-581sin(87)-(-570), 581cos(87)-0>
=< -10.2, 30.4>
Magnitude = sqrt((-10.2)^2+30.4^2)=32.07
direction = atan(30.4/(-10.2)) = 90-71.45=18.55 degrees west of north.