The position vectors of the sides of triangle are `3hati+4hatj+5hatk, hati+7hatk` and `5hati+5hatk`. The distance between the circumcentre and the ortho centre is

Options

A. 0
B. `(3sqrt(274))/(sqrt(11))`
C. `sqrt(306)`
D. `(3)/(2)sqrt(306)`

Answer

To find the distance between the circumcenter and the orthocenter of the triangle with given position vectors, we can follow these steps:

### Step 1: Define the position vectors
Let the position vectors of the vertices of the triangle be:
- \( \vec{A} = 3\hat{i} + 4\hat{j} + 5\hat{k} \)
- \( \vec{B} = \hat{i} + 7\hat{k} \)
- \( \vec{C} = 5\hat{i} + 5\hat{k} \)

### Step 2: Calculate the magnitudes of the position vectors
We can calculate the magnitudes of these vectors to determine their distances from the origin.

1. For \( \vec{A} \):
\[ |\vec{A}| = \sqrt{(3)^2 + (4)^2 + (5)^2} = \sqrt{9 + 16 + 25} = \sqrt{50} \]

2. For \( \vec{B} \):
\[ |\vec{B}| = \sqrt{(1)^2 + (0)^2 + (7)^2} = \sqrt{1 + 0 + 49} = \sqrt{50} \]

3. For \( \vec{C} \):
\[ |\vec{C}| = \sqrt{(5)^2 + (0)^2 + (5)^2} = \sqrt{25 + 0 + 25} = \sqrt{50} \]

### Step 3: Identify the circumcenter
Since all three vertices are equidistant from the origin, the circumcenter \( O \) of the triangle is at the origin:
\[ O = (0, 0, 0) \]

### Step 4: Calculate the centroid of the triangle
The centroid \( G \) of the triangle is given by the formula:
\[ \vec{G} = \frac{\vec{A} + \vec{B} + \vec{C}}{3} \]

Calculating \( \vec{G} \):
\[ \vec{G} = \frac{(3\hat{i} + 4\hat{j} + 5\hat{k}) + (\hat{i} + 0\hat{j} + 7\hat{k}) + (5\hat{i} + 0\hat{j} + 5\hat{k})}{3} \]
\[ = \frac{(3 + 1 + 5)\hat{i} + (4 + 0 + 0)\hat{j} + (5 + 7 + 5)\hat{k}}{3} \]
\[ = \frac{9\hat{i} + 4\hat{j} + 17\hat{k}}{3} \]
\[ = 3\hat{i} + \frac{4}{3}\hat{j} + \frac{17}{3}\hat{k} \]

### Step 5: Calculate the distance between circumcenter and centroid
The distance \( d \) between the circumcenter \( O \) and the centroid \( G \) is given by:
\[ d = |\vec{G} - \vec{O}| = |\vec{G}| \]
Calculating \( |\vec{G}| \):
\[ |\vec{G}| = \sqrt{\left(3\right)^2 + \left(\frac{4}{3}\right)^2 + \left(\frac{17}{3}\right)^2} \]
\[ = \sqrt{9 + \frac{16}{9} + \frac{289}{9}} = \sqrt{9 + \frac{305}{9}} = \sqrt{\frac{81 + 305}{9}} = \sqrt{\frac{386}{9}} = \frac{\sqrt{386}}{3} \]

### Step 6: Calculate the distance between centroid and orthocenter
The distance between the centroid \( G \) and the orthocenter \( H \) is given by:
\[ d_{GH} = 3 \times d_{OG} = 3 \times \frac{\sqrt{386}}{3} = \sqrt{386} \]

### Final Answer
The distance between the circumcenter and the orthocenter is:
\[ \sqrt{386} \approx 19.23 \]