The linear velocity of rotating body is given by `vecv = vecomega xx vecr` where `vecomega` is the angular velocity and `vecr` is the radius vector. The angular velocity of body `vecomega = hati - 2hatj + 2hatk` and this radius vector `vecr = 4hatj - 3hatk`, then `|vecv|` is

Options

A. `sqrt(29)` units
B. `sqrt(31)` units
C. `sqrt(37)` units
D. `sqrt(41)` units

Explanation

To solve the problem, we need to find the magnitude of the linear velocity vector \(\vec{v}\) given by the cross product of the angular velocity vector \(\vec{\omega}\) and the radius vector \(\vec{r}\).

### Step-by-Step Solution:

1. **Identify the vectors:**
- The angular velocity vector is given as:
\[ \vec{\omega} = \hat{i} - 2\hat{j} + 2\hat{k} \]
- The radius vector is given as:
\[ \vec{r} = 4\hat{j} - 3\hat{k} \]

2. **Write the radius vector in full form:**
- Since there is no \(\hat{i}\) component in \(\vec{r}\), we can represent it as:
\[ \vec{r} = 0\hat{i} + 4\hat{j} - 3\hat{k} \]

3. **Calculate the cross product \(\vec{v} = \vec{\omega} \times \vec{r}\):**
- We can set up the determinant to calculate the cross product:
\[ \vec{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & -2 & 2 \\ 0 & 4 & -3 \end{vmatrix} \]

4. **Calculate the determinant:**
- Expanding the determinant:
\[ \vec{v} = \hat{i} \begin{vmatrix} -2 & 2 \\ 4 & -3 \end{vmatrix} - \hat{j} \begin{vmatrix} 1 & 2 \\ 0 & -3 \end{vmatrix} + \hat{k} \begin{vmatrix} 1 & -2 \\ 0 & 4 \end{vmatrix} \]

- Calculating each of the 2x2 determinants:
- For \(\hat{i}\):
\[ (-2)(-3) - (2)(4) = 6 - 8 = -2 \]
- For \(\hat{j}\):
\[ (1)(-3) - (2)(0) = -3 - 0 = -3 \]
- For \(\hat{k}\):
\[ (1)(4) - (-2)(0) = 4 - 0 = 4 \]

- Putting it all together:
\[ \vec{v} = -2\hat{i} + 3\hat{j} + 4\hat{k} \]

5. **Calculate the magnitude of \(\vec{v}\):**
- The magnitude \(|\vec{v}|\) is given by:
\[ |\vec{v}| = \sqrt{(-2)^2 + (3)^2 + (4)^2} \]
- Calculating each term:
\[ |\vec{v}| = \sqrt{4 + 9 + 16} = \sqrt{29} \]

6. **Final Result:**
- Therefore, the magnitude of the linear velocity vector is:
\[ |\vec{v}| = \sqrt{29} \]