The perpendicular distance between the line `vecr = 2hati-2hatj+3hatk+lambda(hati-hatj+4hatk)` and the plane `vecr.(hati + 5hatj + hatk) = 5` is :

Options

A. `(10)/(3sqrt3)`
B. `(10)/(9)`
C. `(10)/(3)`
D. `(3)/(10)`

Explanation

To find the perpendicular distance between the given line and the plane, we can follow these steps:

### Step 1: Identify the components of the line and the plane
The line is given by the vector equation:
\[ \vec{r} = 2\hat{i} - 2\hat{j} + 3\hat{k} + \lambda(\hat{i} - \hat{j} + 4\hat{k}) \]
From this, we can identify:
- Point \( A = 2\hat{i} - 2\hat{j} + 3\hat{k} \)
- Direction vector \( \vec{b} = \hat{i} - \hat{j} + 4\hat{k} \)

The plane is given by the equation:
\[ \vec{r} \cdot (\hat{i} + 5\hat{j} + \hat{k}) = 5 \]
From this, we can identify:
- Normal vector \( \vec{n} = \hat{i} + 5\hat{j} + \hat{k} \)
- Constant \( p = 5 \)

### Step 2: Check if the line is parallel to the plane
To check if the line is parallel to the plane, we calculate the dot product of the direction vector \( \vec{b} \) and the normal vector \( \vec{n} \):
\[ \vec{b} \cdot \vec{n} = (1)(1) + (-1)(5) + (4)(1) = 1 - 5 + 4 = 0 \]
Since the dot product is zero, the line is indeed parallel to the plane.

### Step 3: Calculate the distance from point A to the plane
The distance \( d \) from a point \( A(x_1, y_1, z_1) \) to the plane \( Ax + By + Cz = D \) is given by the formula:
\[ d = \frac{|Ax_1 + By_1 + Cz_1 - D|}{\sqrt{A^2 + B^2 + C^2}} \]
For our case:
- \( A = 1, B = 5, C = 1, D = 5 \)
- Coordinates of point \( A \) are \( (2, -2, 3) \)

Substituting these values into the formula:
\[ d = \frac{|1(2) + 5(-2) + 1(3) - 5|}{\sqrt{1^2 + 5^2 + 1^2}} \]
Calculating the numerator:
\[ = |2 - 10 + 3 - 5| = |2 - 10 + 3 - 5| = |-10| = 10 \]

Calculating the denominator:
\[ = \sqrt{1 + 25 + 1} = \sqrt{27} = 3\sqrt{3} \]

### Step 4: Final distance calculation
Thus, the distance \( d \) is:
\[ d = \frac{10}{3\sqrt{3}} \]

### Final Answer
The perpendicular distance between the line and the plane is:
\[ \frac{10}{3\sqrt{3}} \]