The velocity of a particle is given by `vec(v) = 2hat(i)-hat(j)+2hat(k)` in `m//s` for time interval `t = 0` to `t = 10` sec. If the distance travelled by the particle in given time interval `10n`, find value of `n` is.

Explanation

To solve the problem, we need to find the distance traveled by a particle given its velocity vector over a specific time interval. Let's break down the steps:

### Step 1: Identify the velocity vector
The velocity vector is given as:
\[ \vec{v} = 2\hat{i} - \hat{j} + 2\hat{k} \quad \text{(in m/s)} \]

### Step 2: Calculate the magnitude of the velocity
To find the distance traveled, we first need the magnitude of the velocity vector:
\[ |\vec{v}| = \sqrt{(2)^2 + (-1)^2 + (2)^2} \]
Calculating this:
\[ |\vec{v}| = \sqrt{4 + 1 + 4} = \sqrt{9} = 3 \, \text{m/s} \]

### Step 3: Calculate the distance traveled over the time interval
The distance \( s \) traveled by the particle can be calculated using the formula:
\[ s = |\vec{v}| \times t \]
where \( t \) is the time interval. Given that \( t = 10 \) seconds:
\[ s = 3 \, \text{m/s} \times 10 \, \text{s} = 30 \, \text{m} \]

### Step 4: Relate the distance to the expression \( 10n \)
We know from the problem that the distance traveled is also expressed as \( 10n \):
\[ 10n = 30 \]

### Step 5: Solve for \( n \)
To find \( n \), we rearrange the equation:
\[ n = \frac{30}{10} = 3 \]

### Final Answer
Thus, the value of \( n \) is:
\[ \boxed{3} \]