Resultant Velocity of Boat:
The resultant velocity of the boat can be found using vector addition. We need to consider the velocities of both the river and the boat to determine the resultant velocity.

Given:
- Velocity of the river = 5 m/s towards east
- Velocity of the boat = 10 m/s (magnitude only, direction unknown)

We can represent the velocity of the river as Vr and the velocity of the boat as Vb.

Vector Addition:
To find the resultant velocity, we need to add the velocities of the river and the boat. Since the boat is crossing the river by the shortest path, the resultant velocity will be the vector sum of the two velocities.

Let's assume the angle between the direction of the boat's velocity and the east direction is θ.

Using vector addition, we can find the resultant velocity (Vr) as follows:

Vr = Vb + Vr

Calculating the Resultant Velocity:
Since we know the magnitude of the river's velocity (5 m/s) and the boat's velocity (10 m/s), we can use trigonometry to find the angle θ and the magnitude of the resultant velocity.

Using the magnitude of the resultant velocity (Vr) and the angle θ, we can find the horizontal component of the resultant velocity (Vr_x) and the vertical component of the resultant velocity (Vr_y) using the following equations:

Vr_x = Vr * cos(θ)
Vr_y = Vr * sin(θ)

Direction of the Boat's Velocity:
The direction of the boat's velocity can be determined by the angle θ. If the boat is crossing the river by the shortest path, the angle θ should be such that the horizontal component of the resultant velocity (Vr_x) is equal to the boat's velocity (10 m/s).

By solving the equation Vr_x = Vr * cos(θ) = Vb, we can find the value of θ.

Explanation:
In this scenario, we have a river flowing towards the east with a velocity of 5 m/s and a boat moving with a velocity of 10 m/s (magnitude only). To find the resultant velocity of the boat when it crosses the river by the shortest path, we need to consider both velocities using vector addition.

By finding the magnitude of the resultant velocity and the angle θ, we can determine the direction of the boat's velocity. The angle θ is crucial in determining the direction as it ensures the horizontal component of the resultant velocity matches the boat's velocity.

Using trigonometric equations, we can find the horizontal and vertical components of the resultant velocity. The direction of the boat's velocity will be determined by the angle θ, which ensures the boat crosses the river by the shortest path.

By following these calculations and considerations, we can determine the resultant velocity and direction of the boat accurately.

Direction = North ..but, i think so resultant velocity of boat must be = 5√5 m/s