a man walks 5 km due North east, then 4 km due 30 degree N of E and finally 60 degree due S of E for 2 km. What is his final displacement with respect to his starting point ?

D = 5km[45o] + 4km[30o] + 2km[-60o].

X = 5*cos45 + 4*cos30 + 2*cos(-60)=8 km.
Y=5*sin45 + 4*sin30 + 2*sin(-60)=3.80km.

tanA = Y/X = 3.80/8 = 0.47544
A = 25.4o

D = 8/cos25.4 = 8.86 km.

To find the final displacement with respect to the starting point, we can use vector addition.

First, let's break down the man's movements into two parts:

1. The initial movement is 5 km due northeast. Since northeast is a combination of north and east directions, we can split this movement into two components: one in the north direction and the other in the east direction.

The north component can be found by multiplying the distance traveled (5 km) by the sine of the angle between the northeast direction and due north (45 degrees in this case). Therefore, the north component is 5 km * sin(45°) = 5 km * √(2)/2 = 5√2/2 km ≈ 3.54 km.

The east component can be found by multiplying the distance traveled (5 km) by the cosine of the angle between the northeast direction and due north (45 degrees). Therefore, the east component is 5 km * cos(45°) = 5 km * √(2)/2 = 5√2/2 km ≈ 3.54 km.

So, the initial movement can be expressed as a vector: 3.54 km north + 3.54 km east.

2. The second movement is 4 km due 30 degrees North of East. Similarly, we can find the north and east components of this movement.

The north component is 4 km * sin(30°) = 4 km * 1/2 = 2 km.

The east component is 4 km * cos(30°) = 4 km * √(3)/2 = 4√3/2 km ≈ 3.46 km.

So, the second movement can be expressed as a vector: 2 km north + 3.46 km east.

3. The final movement is 2 km at a 60-degree angle due south of east. Again, we can find the north and east components.

The north component is 2 km * sin(60°) = 2 km * √(3)/2 = √(3) km ≈ 1.73 km.

The east component is 2 km * cos(60°) = 2 km * 1/2 = 1 km.

So, the final movement can be expressed as a vector: 1.73 km south + 1 km east.

To find the total displacement, we can add all the components together:

(3.54 km north + 3.54 km east) + (2 km north + 3.46 km east) + (1.73 km south + 1 km east)

Combining the north components, we get 3.54 km + 2 km - 1.73 km = 3.81 km.

Combining the east components, we get 3.54 km + 3.46 km + 1 km = 8 km.

Therefore, the final displacement with respect to the starting point is approximately 3.81 km north + 8 km east.