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.
The speed with the rain drops hit the person is `:`

Options

A. `10m//s`
B. `10sqrt(2)m//s`
C. `sqrt(180)m//s`
D. `sqrt(360)m//s`

Explanation

To find the speed with which the raindrops hit the person, we need to calculate the relative velocity of the rain with respect to the person. Here’s a step-by-step solution:

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

### Step 2: Calculate the relative velocity of rain with respect to the person
The relative velocity of the rain with respect to the person is given by:
\[ \vec{V}_{\text{rain, relative}} = \vec{V}_{\text{rain}} - \vec{V}_{\text{person}} \]

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

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

So, the relative velocity becomes:
\[ \vec{V}_{\text{rain, relative}} = -10 \hat{i} + 0 \hat{j} - 10 \hat{k} \]

### Step 4: Calculate the magnitude of the relative velocity
The magnitude of the relative velocity is given by:
\[ |\vec{V}_{\text{rain, relative}}| = \sqrt{(-10)^2 + 0^2 + (-10)^2} \]
Calculating this gives:
\[ |\vec{V}_{\text{rain, relative}}| = \sqrt{100 + 0 + 100} = \sqrt{200} = 10\sqrt{2} \]

### Final Answer
The speed with which the raindrops hit the person is:
\[ 10\sqrt{2} \text{ m/s} \]

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