Cities P,Q,R,S are such that Q is 4km due east of P,R is 3km due north of Q and S is 4km S 50degree W from R. Calculate the distance and bearing of P from S short and straight.
Using vectors:
PS = 4(cos0,sin0) + 3(cos90°,sin90°) + 4(cos220°,sin220°)
= (4,0) + (0,3) + (-3.0642, -2.5712)
= (.9358, .4288)
magnitude = √(.9358^2 + .4288^2) = appr 1.029
angle = tan^-1 (.4288/.9358) = 24.62°
or using your notation: N 65.38° E
All angles are measured CCW from +x-axis.
Given:
PQ = 4km[0o].
QR = 3km[90o].
RS = 4km[50o] W. of S. = 4km[220o] CCW.
SP = ?.
SP = 4[0o] + 3[90] + 4[220o].
X = 4*Cos0 + 3*Cos90 + 4*Cos220 = 4 + 0 - 3.06 = 0.94 km.
Y = 4*sin0 + 3*sin90 + 4*sin220 = 0 + 3 - 2.57 = 0.43 km.
SP = 0.94 + 0.43i = 1.03km[24.6o] CCW.
To calculate the distance and bearing of P from S in a short and straight manner, we can use the concept of vector addition.
Step 1: Represent the distances and directions as vectors:
Let the vector from P to Q be denoted as PQ, the vector from Q to R be denoted as QR, and the vector from R to S be denoted as RS.
PQ = 4 km due east (represented as 4i)
QR = 3 km due north (represented as 3j)
RS = 4 km S 50° W (represented as -4 cos(50°)i - 4 sin(50°)j)
Step 2: Use vector addition to find the vector from P to S (PS):
PS = PQ + QR + RS
PS = 4i + 3j + (-4 cos(50°)i - 4 sin(50°)j)
Step 3: Simplify the vector PS:
PS = (4 - 4 cos(50°))i + (3 - 4 sin(50°))j
Step 4: Calculate the magnitude (distance) of the vector PS:
Magnitude of PS = √[(4 - 4 cos(50°))^2 + (3 - 4 sin(50°))^2] km
Step 5: Calculate the bearing of P from S:
Bearing of P from S = tan^(-1)[(3 - 4 sin(50°)) / (4 - 4 cos(50°))] degrees
Step 6: Substitute in the numerical values and calculate:
- Magnitude of PS = √[(4 - 4 cos(50°))^2 + (3 - 4 sin(50°))^2] km
- Bearing of P from S = tan^(-1)[(3 - 4 sin(50°)) / (4 - 4 cos(50°))] degrees
Solving these equations will give you the numerical values for the distance and bearing of P from S in a short and straight line.
To calculate the distance and bearing of P from S, we can use the concept of vector addition.
Step 1: Draw a diagram
Let's draw a diagram to visualize the given information:
```
R
|
| \
| \
| Q
S|———P
```
Step 2: Calculate the coordinates of each city
Start by assuming the coordinates of city P as (0,0).
Since Q is 4km due east of P, the coordinates of Q would be (4,0).
Next, as R is 3km due north of Q, the coordinates of R would be (4,3).
Finally, as S is 4km S 50° W from R, we need to convert the bearing to coordinates. We can calculate the difference in coordinates based on the distance and angle:
```
Δx = 4km * cos(50°)
Δy = -4km * sin(50°)
```
Calculating these values, we find that S has coordinates (4 - 4 * cos(50°), 3 - 4 * sin(50°)).
Step 3: Calculate the distance between P and S
We can calculate the distance between two points (x1, y1) and (x2, y2) using the distance formula:
```
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
```
Calculating the distance, we get:
```
distance = sqrt((4 - 4 * cos(50°))^2 + (3 - 4 * sin(50°))^2)
```
Step 4: Calculate the bearing of P from S
We can calculate the bearing using the tangent of an angle:
```
bearing = arctan((y2 - y1) / (x2 - x1))
```
Calculating the bearing, we get:
```
bearing = arctan((3 - 4 * sin(50°)) / (4 - 4 * cos(50°)))
```
Step 5: Calculate the final values
Using a calculator to compute the distance and bearing, we obtain:
```
distance ≈ 2.958 km
bearing ≈ -25.58°
```
Therefore, the distance from P to S, taking the shortest straight path, is approximately 2.958 km. The bearing of P from S is approximately -25.58°. Note: The negative sign indicates that the bearing is in the West direction.