A ship sails 15.0 mi on a course S40°10’W and then 21.0 mi on a course N28°20’W. Find the distance and direction of the last position from the first.
convert to x-y, add the components, and convert back to compass bearing.
15 @ S40°10’W = <-9.675,-11.463>
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To find the distance and direction of the last position from the first, we can use the concept of vector addition.
Step 1: Convert the compass bearings to decimal degrees.
S40°10’W = 40.167°
N28°20’W = -28.333°
Step 2: Represent the first position as a vector. Let's call it Vector A.
The magnitude of Vector A is 15.0 miles, and its direction is 40.167° west of south.
Step 3: Represent the second position as a vector. Let's call it Vector B.
The magnitude of Vector B is 21.0 miles, and its direction is 28.333° west of north.
Step 4: Determine the x and y components of Vector A and Vector B.
Vector A: x = 15.0 * sin(40.167°), y = -15.0 * cos(40.167°)
Vector B: x = -21.0 * sin(28.333°), y = 21.0 * cos(28.333°)
Step 5: Add the x and y components of Vector A and Vector B to find the resultant vector.
Resultant vector (Vector R) = (Ax + Bx) i + (Ay + By) j
Step 6: Calculate the magnitude (distance) and direction of Vector R.
Magnitude of Vector R = sqrt((Ax + Bx)^2 + (Ay + By)^2)
Direction of Vector R = arctan((Ay + By) / (Ax + Bx))
By following these steps, the distance and direction of the last position from the first can be determined.
To find the distance and direction of the last position from the first, we can use vector addition.
1. Convert the given directions into vectors:
a. S40°10’W can be represented as a vector with a magnitude of 15.0 mi and a direction 40°10’ west of south.
b. N28°20’W can be represented as a vector with a magnitude of 21.0 mi and a direction 28°20’ west of north.
2. Convert the directions to standard notation:
a. S40°10’W = 180° - (90° + 40°10’) = 180° - 130°10’ = 49°50’ (west of south)
b. N28°20’W = 180° + 28°20’ = 208°20’ (west of north)
3. Convert the directions to radians:
a. 49°50’ = (49 + 50/60)° = (49 + 5/6)° = (49 * π/180 + 5/6 * π/180) rad
b. 208°20’ = (208 + 20/60)° = (208 + 1/3)° = (208 * π/180 + 1/3 * π/180) rad
4. Convert the directions to Cartesian coordinates:
a. S40°10’W = (-15.0 * sin(49 * π/180 + 5/6 * π/180), -15.0 * cos(49 * π/180 + 5/6 * π/180))
b. N28°20’W = (21.0 * sin(208 * π/180 + 1/3 * π/180), -21.0 * cos(208 * π/180 + 1/3 * π/180))
5. Perform vector addition:
The position vector from the first point is the sum of the two vectors calculated above:
Position vector = (-15.0 * sin(49 * π/180 + 5/6 * π/180) + 21.0 * sin(208 * π/180 + 1/3 * π/180), -15.0 * cos(49 * π/180 + 5/6 * π/180) - 21.0 * cos(208 * π/180 + 1/3 * π/180))
6. Calculate the magnitude of the position vector:
Magnitude = sqrt(x^2 + y^2), where x and y are the components of the position vector.
7. Calculate the direction of the position vector:
Direction = atan2(y, x), where x and y are the components of the position vector.
By following these steps, you should be able to find the distance and direction of the last position from the first.