A man is walking with `3 m//s,` due east. Rain is falling vetically downwards with speed `4 m//s.` Find the direction in which man should hold his umbrella, so that rain does not wet him.

Explanation

To solve the problem of determining the direction in which the man should hold his umbrella so that he does not get wet from the rain, we will follow these steps:

### Step 1: Understand the velocities involved
- The man is walking due east with a velocity \( V_m = 3 \, \text{m/s} \).
- The rain is falling vertically downwards with a velocity \( V_r = 4 \, \text{m/s} \).

### Step 2: Determine the relative velocity of the rain with respect to the man
To find the direction in which the man should hold his umbrella, we need to calculate the relative velocity of the rain with respect to the man. This can be found using vector subtraction:

\[ \text{Velocity of rain with respect to man} = \text{Velocity of rain} - \text{Velocity of man} \]

Since the rain is falling straight down and the man is moving horizontally, we can represent the velocities as vectors:
- Velocity of rain: \( \vec{V_r} = (0, -4) \) (where the negative sign indicates downward direction)
- Velocity of man: \( \vec{V_m} = (3, 0) \) (where the positive x-direction indicates east)

Now, we can find the relative velocity:

\[ \vec{V_{rm}} = \vec{V_r} - \vec{V_m} = (0, -4) - (3, 0) = (-3, -4) \]

### Step 3: Calculate the angle at which to hold the umbrella
To find the direction in which the man should hold the umbrella, we need to determine the angle \( \theta \) that the relative velocity vector makes with the vertical. We can use the tangent function:

\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{V_m}{V_r} = \frac{3}{4} \]

### Step 4: Find the angle using the inverse tangent function
Now, we can calculate \( \theta \):

\[ \theta = \tan^{-1}\left(\frac{3}{4}\right) \]

Using a calculator, we find:

\[ \theta \approx 36.87^\circ \]

### Step 5: Conclusion
The man should hold his umbrella at an angle of approximately \( 37^\circ \) from the vertical towards the east to avoid getting wet from the rain.