Find the value of 'm' for which the line `vec(r) = ( hati + 2 hatk ) + lambda (2 hati - m hatj - 3 hatk)` is parallel to the plane
`vec(r).(m hati + 3 hatj + hatk )` = 4.

Answer

To find the value of \( m \) for which the line

\[ \vec{r} = \hat{i} + 2\hat{k} + \lambda(2\hat{i} - m\hat{j} - 3\hat{k}) \]

is parallel to the plane given by

\[ \vec{r} \cdot (m\hat{i} + 3\hat{j} + \hat{k}) = 4, \]

we will follow these steps:

### Step 1: Identify the direction vector of the line

The direction vector of the line can be extracted from the equation:

\[ \vec{d} = 2\hat{i} - m\hat{j} - 3\hat{k}. \]

### Step 2: Identify the normal vector of the plane

The normal vector of the plane can be identified from the equation of the plane:

\[ \vec{n} = m\hat{i} + 3\hat{j} + \hat{k}. \]

### Step 3: Set the condition for parallelism

For the line to be parallel to the plane, the direction vector of the line must be orthogonal to the normal vector of the plane. This means that their dot product must be zero:

\[ \vec{d} \cdot \vec{n} = 0. \]

### Step 4: Compute the dot product

Calculating the dot product:

\[ (2\hat{i} - m\hat{j} - 3\hat{k}) \cdot (m\hat{i} + 3\hat{j} + \hat{k}) = 0. \]

Expanding this, we get:

\[ 2m + (-m)(3) + (-3)(1) = 0. \]

This simplifies to:

\[ 2m - 3m - 3 = 0. \]

### Step 5: Solve for \( m \)

Combining like terms gives:

\[ -m - 3 = 0. \]

Adding 3 to both sides:

\[ -m = 3. \]

Multiplying both sides by -1 gives:

\[ m = -3. \]

### Conclusion

Thus, the value of \( m \) for which the line is parallel to the plane is:

\[ \boxed{-3}. \]

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