The linear velocity of a rotating body is given by `vec(v)= vec(omega)xxvec(r )`, where `vec(omega)` is the angular velocity and `vec(r )` is the radius vector. The angular velocity of a body is `vec(omega)= hat(i)-2hat(j)+2hat(k)` and the radius vector `vec(r )= 4hat(j)-3hat(k)`, then `|vec(v)|` 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 linear velocity vector \(\vec{v}\) using the cross product of the angular velocity vector \(\vec{\omega}\) and the radius vector \(\vec{r}\). The given vectors are:

\[ \vec{\omega} = \hat{i} - 2\hat{j} + 2\hat{k} \]
\[ \vec{r} = 4\hat{j} - 3\hat{k} \]

### Step 1: Set up the cross product

The cross product \(\vec{v} = \vec{\omega} \times \vec{r}\) can be calculated using the determinant of a 3x3 matrix:

\[ \vec{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & -2 & 2 \\ 0 & 4 & -3 \end{vmatrix} \]

### Step 2: Calculate the determinant

To compute the determinant, we expand it as follows:

\[ \vec{v} = \hat{i} \begin{vmatrix} -2 & 2 \\ 4 & -3 \end{vmatrix} - \hat{j} \begin{vmatrix} 1 & 2 \\ 0 & -3 \end{vmatrix} + \hat{k} \begin{vmatrix} 1 & -2 \\ 0 & 4 \end{vmatrix} \]

Calculating each of these 2x2 determinants:

1. For \(\hat{i}\):
\[ \begin{vmatrix} -2 & 2 \\ 4 & -3 \end{vmatrix} = (-2)(-3) - (2)(4) = 6 - 8 = -2 \]

2. For \(\hat{j}\):
\[ \begin{vmatrix} 1 & 2 \\ 0 & -3 \end{vmatrix} = (1)(-3) - (2)(0) = -3 - 0 = -3 \]

3. For \(\hat{k}\):
\[ \begin{vmatrix} 1 & -2 \\ 0 & 4 \end{vmatrix} = (1)(4) - (-2)(0) = 4 - 0 = 4 \]

Putting it all together, we have:

\[ \vec{v} = -2\hat{i} - (-3)\hat{j} + 4\hat{k} = -2\hat{i} + 3\hat{j} + 4\hat{k} \]

### Step 3: Find the magnitude of \(\vec{v}\)

The magnitude of the vector \(\vec{v}\) is given by:

\[ |\vec{v}| = \sqrt{(-2)^2 + 3^2 + 4^2} \]

Calculating each term:

\[ (-2)^2 = 4, \quad 3^2 = 9, \quad 4^2 = 16 \]

Now summing these:

\[ |\vec{v}| = \sqrt{4 + 9 + 16} = \sqrt{29} \]

### Final Answer

Thus, the magnitude of the linear velocity \(|\vec{v}|\) is:

\[ |\vec{v}| = \sqrt{29} \text{ units} \]