Find the distance of the point with position vector `- hati - 5 hatj - 10 hatk ` from the point of intersection of the line `vec(r) = (2 hati - hatj + 2 hatk) + lambda ( 3 hati + 4 hatj + 12 hatk) ` with the plane `vec(r). (hati - hatj + hatk)` = 5 .

Answer

To find the distance of the point with position vector \(-\hat{i} - 5\hat{j} - 10\hat{k}\) from the point of intersection of the given line and plane, we will follow these steps:

### Step 1: Identify the line and plane equations
The line is given in vector form as:
\[ \vec{r} = (2\hat{i} - \hat{j} + 2\hat{k}) + \lambda(3\hat{i} + 4\hat{j} + 12\hat{k}) \]
The plane is given by the equation:
\[ \vec{r} \cdot (\hat{i} - \hat{j} + \hat{k}) = 5 \]

### Step 2: Convert the line equation to parametric form
From the line equation, we can express the coordinates \(x\), \(y\), and \(z\) in terms of \(\lambda\):
- \(x = 2 + 3\lambda\)
- \(y = -1 + 4\lambda\)
- \(z = 2 + 12\lambda\)

### Step 3: Substitute into the plane equation
Substituting these expressions into the plane equation:
\[ (2 + 3\lambda) - (-1 + 4\lambda) + (2 + 12\lambda) = 5 \]
This simplifies to:
\[ 2 + 3\lambda + 1 - 4\lambda + 2 + 12\lambda = 5 \]
Combining like terms gives:
\[ (3\lambda - 4\lambda + 12\lambda) + (2 + 1 + 2) = 5 \]
\[ 11\lambda + 5 = 5 \]

### Step 4: Solve for \(\lambda\)
Now, we can solve for \(\lambda\):
\[ 11\lambda = 5 - 5 \]
\[ 11\lambda = 0 \implies \lambda = 0 \]

### Step 5: Find the coordinates of the intersection point
Substituting \(\lambda = 0\) back into the parametric equations gives:
- \(x = 2 + 3(0) = 2\)
- \(y = -1 + 4(0) = -1\)
- \(z = 2 + 12(0) = 2\)

Thus, the point of intersection is \((2, -1, 2)\).

### Step 6: Calculate the distance from the given point
The given point has coordinates \((-1, -5, -10)\). The distance \(d\) between the points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
Substituting the coordinates:
\[ d = \sqrt{(2 - (-1))^2 + (-1 - (-5))^2 + (2 - (-10))^2} \]
\[ = \sqrt{(2 + 1)^2 + (-1 + 5)^2 + (2 + 10)^2} \]
\[ = \sqrt{3^2 + 4^2 + 12^2} \]
\[ = \sqrt{9 + 16 + 144} \]
\[ = \sqrt{169} = 13 \]

### Final Answer
The distance from the point \(-\hat{i} - 5\hat{j} - 10\hat{k}\) to the point of intersection is \(13\).

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