Given information:
A person is crossing a road with a certain velocity due north. The person sees a car moving towards the east. The relative velocity of the car with respect to the person is √2 times that of the velocity of the person.

To find:
The angle made by the relative velocity with the east.

Explanation:
1. Understanding relative velocity:
Relative velocity is the velocity of an object with respect to another object. It is the difference between the velocities of the two objects.

2. Finding the relative velocity:
In this scenario, the person is moving north, and the car is moving east. To find the relative velocity of the car with respect to the person, we need to subtract the velocity of the person from the velocity of the car.

3. Setting up coordinate axes:
We can set up a coordinate system with the x-axis pointing east and the y-axis pointing north. The person's velocity is in the positive y-direction, and the car's velocity is in the positive x-direction.

4. Representing velocities as vectors:
The person's velocity can be represented as a vector Vp in the positive y-direction. The car's velocity can be represented as a vector Vc in the positive x-direction.

5. Calculating the magnitude of relative velocity:
Given that the relative velocity of the car with respect to the person is √2 times that of the person's velocity, we can write it as:
|Vr| = √2 * |Vp|

6. Calculating the angle:
To find the angle made by the relative velocity with the east, we can use trigonometry. The angle θ can be calculated using the equation:
tan(θ) = |Vp| / |Vc|

7. Substituting values:
We know that |Vr| = √2 * |Vp| and |Vp| / |Vc| = tan(θ). Substituting these values into the equation, we get:
tan(θ) = (√2 * |Vp|) / |Vc|

8. Solving for θ:
To find the value of θ, we can take the inverse tangent (arctan) of both sides of the equation:
θ = arctan[(√2 * |Vp|) / |Vc|]

9. Conclusion:
The angle made by the relative velocity with the east is given by the equation θ = arctan[(√2 * |Vp|) / |Vc|]. By substituting the known values of the person's and car's velocities, you can calculate the angle.