The distance between the line `r=2hat(i)-2hat(j)+3hat(k)+lambda(hat(i)-hat(j)+4hat(k))` and the plane `rcdot(hat(i)+5hat(j)+hat(k))=5,` is

Options

A. `(10)/(9)`
B. `(10)/(3sqrt(3))`
C. `(10)/(3)`
D. None of these

Answer

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

### Step 1: Identify the line and the plane equations
The line is given by:
\[ \mathbf{r} = (2\hat{i} - 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} - \hat{j} + 4\hat{k}) \]
This can be rewritten as:
\[ \mathbf{r} = (2 + \lambda)\hat{i} + (-2 - \lambda)\hat{j} + (3 + 4\lambda)\hat{k} \]

The plane is given by the equation:
\[ \mathbf{r} \cdot (\hat{i} + 5\hat{j} + \hat{k}) = 5 \]
This can be expressed as:
\[ x + 5y + z = 5 \]

### Step 2: Check if the line is parallel to the plane
The direction vector of the line is:
\[ \mathbf{d} = \hat{i} - \hat{j} + 4\hat{k} \]
The normal vector of the plane is:
\[ \mathbf{n} = \hat{i} + 5\hat{j} + \hat{k} \]
To check if the line is parallel to the plane, we need to see if \(\mathbf{d} \cdot \mathbf{n} = 0\):
\[ \mathbf{d} \cdot \mathbf{n} = (1)(1) + (-1)(5) + (4)(1) = 1 - 5 + 4 = 0 \]
Since the dot product is zero, the line is indeed parallel to the plane.

### Step 3: Find a point on the line
To find a specific point on the line, we can set \(\lambda = 0\):
\[ \text{Point on the line} = (2, -2, 3) \]

### Step 4: Calculate the distance from the point to the plane
We can use the formula for the distance \(D\) from a point \((x_0, y_0, z_0)\) to the plane \(Ax + By + Cz + D = 0\):
\[ D = \frac{|Ax_0 + By_0 + Cz_0 + D|}{\sqrt{A^2 + B^2 + C^2}} \]
For our plane \(x + 5y + z - 5 = 0\), we have:
- \(A = 1\)
- \(B = 5\)
- \(C = 1\)
- \(D = -5\)

Substituting the point \((2, -2, 3)\):
\[ D = \frac{|1(2) + 5(-2) + 1(3) - 5|}{\sqrt{1^2 + 5^2 + 1^2}} = \frac{|2 - 10 + 3 - 5|}{\sqrt{1 + 25 + 1}} = \frac{|-10|}{\sqrt{27}} = \frac{10}{\sqrt{27}} = \frac{10}{3\sqrt{3}} \]

### Step 5: Simplify the distance
To express the distance in a more standard form:
\[ D = \frac{10\sqrt{3}}{9} \]

### Final Answer
The distance between the line and the plane is:
\[ \frac{10\sqrt{3}}{9} \]