If `p^(th), q^(th), r^(th)`, terms of a G.P. are the positive numbers a, b, c respectively then angle between the vectors `log a^3 hat i + log b^3 hatj + log c^3 hatk` and `(q - r)hati + (r - p)hatj + (p - q) hatk` is:

Options

A. `pi/2`
B. `pi/3`
C. `0`
D. `sin^(-1)""(1)/(sqrt(p^2 + q^2 + r^2))`

Explanation

To solve the problem, we need to find the angle between two vectors given in the question. Let's break it down step by step.

### Step 1: Identify the Terms of the G.P.
Given that the \( p^{th}, q^{th}, r^{th} \) terms of a G.P. are positive numbers \( a, b, c \) respectively, we can express these terms in terms of the first term \( m \) and the common ratio \( n \):
- \( a = m \cdot n^{p-1} \)
- \( b = m \cdot n^{q-1} \)
- \( c = m \cdot n^{r-1} \)

### Step 2: Express the Vectors
We need to express the vectors based on the logarithms of \( a, b, c \):
- The first vector is \( \mathbf{A} = \log(a^3) \hat{i} + \log(b^3) \hat{j} + \log(c^3) \hat{k} \).
- Using the properties of logarithms, we can rewrite this as:
\[ \mathbf{A} = 3\log(a) \hat{i} + 3\log(b) \hat{j} + 3\log(c) \hat{k} \]
\[ = 3\left(\log(m) + (p-1)\log(n)\right) \hat{i} + 3\left(\log(m) + (q-1)\log(n)\right) \hat{j} + 3\left(\log(m) + (r-1)\log(n)\right) \hat{k} \]

### Step 3: Simplify the First Vector
Now, we can factor out the common terms:
\[ \mathbf{A} = 3\log(m) \hat{i} + 3\log(m) \hat{j} + 3\log(m) \hat{k} + 3(p-1)\log(n) \hat{i} + 3(q-1)\log(n) \hat{j} + 3(r-1)\log(n) \hat{k} \]
\[ = 3\log(m) \left(\hat{i} + \hat{j} + \hat{k}\right) + 3\log(n) \left((p-1) \hat{i} + (q-1) \hat{j} + (r-1) \hat{k}\right) \]

### Step 4: Define the Second Vector
The second vector given is:
\[ \mathbf{B} = (q - r) \hat{i} + (r - p) \hat{j} + (p - q) \hat{k} \]

### Step 5: Find the Dot Product
To find the angle \( \theta \) between the two vectors, we use the dot product:
\[ \mathbf{A} \cdot \mathbf{B} = \left(3\log(m) + 3\log(n)(p-1)\right)(q - r) + \left(3\log(m) + 3\log(n)(q-1)\right)(r - p) + \left(3\log(m) + 3\log(n)(r-1)\right)(p - q) \]

### Step 6: Calculate Magnitudes
Next, we calculate the magnitudes of both vectors:
\[ |\mathbf{A}| = \sqrt{(3\log(m) + 3\log(n)(p-1))^2 + (3\log(m) + 3\log(n)(q-1))^2 + (3\log(m) + 3\log(n)(r-1))^2} \]
\[ |\mathbf{B}| = \sqrt{(q - r)^2 + (r - p)^2 + (p - q)^2} \]

### Step 7: Use the Cosine Formula
Using the dot product and magnitudes, we have:
\[ \cos(\theta) = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}| |\mathbf{B}|} \]

### Step 8: Solve for the Angle
Finally, we can find the angle \( \theta \) using:
\[ \theta = \cos^{-1}\left(\frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}| |\mathbf{B}|}\right) \]