Rain is falling with a velocity `(-4hat(i)+8hat(j)-10hat(k))`. A person is moving with a velocity of `(6hat(i)+8hat(j))` on the ground.
Find the velocity of rain with respect to man and the direction from which the rain appears to be coming.

Explanation

To find the velocity of rain with respect to the man and the direction from which the rain appears to be coming, we can follow these steps:

### Step 1: Identify the given velocities
- The velocity of rain \( \vec{V}_{\text{rain}} \) is given as:
\[ \vec{V}_{\text{rain}} = -4\hat{i} + 8\hat{j} - 10\hat{k} \]
- The velocity of the man \( \vec{V}_{\text{man}} \) is given as:
\[ \vec{V}_{\text{man}} = 6\hat{i} + 8\hat{j} \]

### Step 2: Calculate the relative velocity of rain with respect to the man
The velocity of rain with respect to the man \( \vec{V}_{\text{rain, man}} \) can be calculated using the formula:
\[ \vec{V}_{\text{rain, man}} = \vec{V}_{\text{rain}} - \vec{V}_{\text{man}} \]

Substituting the values:
\[ \vec{V}_{\text{rain, man}} = (-4\hat{i} + 8\hat{j} - 10\hat{k}) - (6\hat{i} + 8\hat{j}) \]

### Step 3: Perform the vector subtraction
Now, we perform the subtraction component-wise:
- For the \( \hat{i} \) component:
\[ -4 - 6 = -10 \]
- For the \( \hat{j} \) component:
\[ 8 - 8 = 0 \]
- For the \( \hat{k} \) component:
\[ -10 - 0 = -10 \]

Thus, we get:
\[ \vec{V}_{\text{rain, man}} = -10\hat{i} + 0\hat{j} - 10\hat{k} \]

### Step 4: Write the final expression for the relative velocity
So, the velocity of rain with respect to the man is:
\[ \vec{V}_{\text{rain, man}} = -10\hat{i} - 10\hat{k} \]

### Step 5: Determine the direction of the rain
To find the direction from which the rain appears to be coming, we can analyze the components of the relative velocity vector:
- The \( \hat{i} \) component is negative, indicating that the rain is coming from the right (in the negative \( \hat{i} \) direction).
- The \( \hat{k} \) component is negative, indicating that the rain is coming from above (in the negative \( \hat{k} \) direction).

### Step 6: Calculate the angle of the rain's direction
To find the angle of the rain's direction, we can use the tangent function:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{10}{10} = 1 \]
Thus,
\[ \theta = \tan^{-1}(1) = 45^\circ \]

### Conclusion
The rain appears to be coming at an angle of \( 45^\circ \) from the horizontal (the \( \hat{i} \) direction).

### Final Answer
The velocity of rain with respect to the man is:
\[ \vec{V}_{\text{rain, man}} = -10\hat{i} - 10\hat{k} \]
And the direction from which the rain appears to be coming is at an angle of \( 45^\circ \) from the horizontal.

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