a car leaves P and moves 20km north to Q from Q it moves 15km on abearing N45 E TO R
use the law of cosines to find the distance PR
If P is at (0,0) the R is at (x,y) = (15/√2,20+15/√2)
So the bearing of R from P is 90-θ, where tanθ = y/x
All angles are measured CW from +y-axis.
PR = PQ+QR = 20km[0o]+15km[45o]
D = (20*sin0+15*sin45)+(20*cos0+15*cos45)i
D = 10.61+30.61i = 32.4km[19.1o].
To determine the final position of the car, we need to break down its movements and calculate the coordinates step by step.
1. Start at point P (let's assume P is the origin of a coordinate system). At this point, the car is located at coordinates (0, 0).
2. The car moves 20 km north from P to Q. Since it's moving directly north, the coordinates change to (0, 20).
3. From point Q, the car proceeds on a bearing of N45°E. This means it is moving northeast. To determine the new coordinates, we need to find the horizontal and vertical components of this movement.
- The horizontal component can be calculated by: distance * cos(angle)
- The vertical component can be calculated by: distance * sin(angle)
Given that the distance is 15 km and the angle is 45°, we can calculate the horizontal and vertical components as follows:
- Horizontal component = 15 km * cos(45°) = 15 km * √(2)/2 ≈ 15 km * 0.707 ≈ 10.606 km
- Vertical component = 15 km * sin(45°) = 15 km * √(2)/2 ≈ 15 km * 0.707 ≈ 10.606 km
4. Adding the horizontal and vertical components to the previous coordinates (Q), we get the new coordinates (R):
- New x-coordinate = Previous x-coordinate + Horizontal component = 0 + 10.606 km ≈ 10.606 km
- New y-coordinate = Previous y-coordinate + Vertical component = 20 + 10.606 km ≈ 30.606 km
Therefore, the final coordinates of point R are approximately (10.606, 30.606).