There is an easy way to do this.
1) Convert all bearings from N as 000, clockwise.
add them as vectors:
R= (46Cos105+22Cos167+14Cos194)North + (46Sin105+22Sin167+14Sin194)East
do the math. check my angles.
1) Convert all bearings from N as 000, clockwise.
add them as vectors:
R= (46Cos105+22Cos167+14Cos194)North + (46Sin105+22Sin167+14Sin194)East
do the math. check my angles.
thankyou
First, let's break down each leg of the journey into its north-south and east-west components:
For the first leg:
Distance = 46 km
Angle = 15 degrees south of east
The north-south (y-component) displacement can be found using the sine of the angle:
Displacement (north-south) = Distance * sin(angle) = 46 km * sin(15 degrees) = 11.92 km south
The east-west (x-component) displacement can be found using the cosine of the angle:
Displacement (east-west) = Distance * cos(angle) = 46 km * cos(15 degrees) = 44.61 km east
For the second leg:
Distance = 22 km
Angle = 13 degrees east of south
The north-south (y-component) displacement can be found using the cosine of the angle (since the angle is measured from the south):
Displacement (north-south) = Distance * cos(angle) = 22 km * cos(13 degrees) = 21.695 km south
The east-west (x-component) displacement can be found using the sine of the angle (since the angle is measured from the south):
Displacement (east-west) = Distance * sin(angle) = 22 km * sin(13 degrees) = 4.84 km east
For the third leg:
Distance = 14 km
Angle = 14 degrees west of south
The north-south (y-component) displacement can be found using the cosine of the angle (since the angle is measured from the south):
Displacement (north-south) = Distance * cos(angle) = 14 km * cos(14 degrees) = 13.325 km south
The east-west (x-component) displacement can be found using the sine of the angle (since the angle is measured from the south):
Displacement (east-west) = Distance * sin(angle) = 14 km * sin(14 degrees) = 3.13 km west
Now, let's add up all the north-south and east-west components separately to find the total north-south and east-west displacements:
Total north-south displacement = Displacement (north-south) from the first leg + Displacement (north-south) from the second leg + Displacement (north-south) from the third leg
= 11.92 km south + 21.695 km south + 13.325 km south = 46.94 km south
Total east-west displacement = Displacement (east-west) from the first leg + Displacement (east-west) from the second leg + Displacement (east-west) from the third leg
= 44.61 km east + 4.84 km east + (-3.13 km west) = 46.32 km east
Now, we can calculate the resultant displacement using the Pythagorean theorem:
Resultant displacement = sqrt((Total north-south displacement)^2 + (Total east-west displacement)^2)
= sqrt((46.94 km)^2 + (46.32 km)^2) = sqrt(4397.63 km^2 + 2143.89 km^2) = sqrt(6538.52 km^2) = 80.93 km
Finally, we can find the direction of the resultant displacement (angle with respect to the north) using trigonometry:
Direction = arctan(Total east-west displacement / Total north-south displacement)
= arctan(46.32 km east / 46.94 km south) = arctan(0.987)
≈ 46.48 degrees south of east
Therefore, the tern's resultant displacement is approximately 80.93 km at an angle of 46.48 degrees south of east.