Suppose `a, b, c gt 0` and are respectively the pth, qth and rth terms of a G.P. Let
`x=(loga)i+(logb)j+(logc)k`
`y=(q-r)i+(r-p)j+(p-q)k`
If angle between x and y is `kpi`, then k = ______

Answer

To solve the problem, we need to find the value of \( k \) such that the angle between the vectors \( x \) and \( y \) is \( k\pi \).

### Step 1: Define the vectors
We are given:
\[ x = (\log a)i + (\log b)j + (\log c)k \]
\[ y = (q - r)i + (r - p)j + (p - q)k \]

### Step 2: Calculate the dot product \( x \cdot y \)
The dot product of two vectors \( x \) and \( y \) is given by:
\[ x \cdot y = (\log a)(q - r) + (\log b)(r - p) + (\log c)(p - q) \]

### Step 3: Simplify the dot product using properties of logarithms
We can rewrite the dot product using properties of logarithms:
\[ x \cdot y = \log a \cdot (q - r) + \log b \cdot (r - p) + \log c \cdot (p - q) \]
Using the property \( \log m + \log n = \log(mn) \), we can express the terms in a different form.

### Step 4: Express terms as powers of \( a, b, c \)
Since \( a, b, c \) are the \( p \)-th, \( q \)-th, and \( r \)-th terms of a geometric progression (G.P.), we can express them in terms of a common ratio \( r \):
\[ a = \alpha r^{p-1}, \quad b = \alpha r^{q-1}, \quad c = \alpha r^{r-1} \]
where \( \alpha \) is the first term of the G.P.

### Step 5: Substitute \( a, b, c \) into the dot product
Substituting these values into the dot product:
\[ x \cdot y = (q - r) \log(\alpha r^{p-1}) + (r - p) \log(\alpha r^{q-1}) + (p - q) \log(\alpha r^{r-1}) \]
This simplifies to:
\[ x \cdot y = (q - r)(\log \alpha + (p-1) \log r) + (r - p)(\log \alpha + (q-1) \log r) + (p - q)(\log \alpha + (r-1) \log r) \]

### Step 6: Combine like terms
By combining the terms, we can factor out \( \log \alpha \) and \( \log r \):
\[ x \cdot y = \log \alpha \cdot [(q - r) + (r - p) + (p - q)] + \log r \cdot [(q - r)(p-1) + (r - p)(q-1) + (p - q)(r-1)] \]
The first bracket simplifies to zero:
\[ (q - r) + (r - p) + (p - q) = 0 \]
Thus, we have:
\[ x \cdot y = \log r \cdot [(q - r)(p-1) + (r - p)(q-1) + (p - q)(r-1)] \]

### Step 7: Analyze the second term
To find the angle between \( x \) and \( y \), we need to check if \( x \cdot y = 0 \):
If \( x \cdot y = 0 \), then \( \cos \theta = 0 \) implies \( \theta = \frac{\pi}{2} \).

### Step 8: Relate \( \theta \) to \( k\pi \)
Since we know \( \theta = k\pi \), we have:
\[ \frac{\pi}{2} = k\pi \]
Thus, solving for \( k \):
\[ k = \frac{1}{2} \]

### Final Answer
The value of \( k \) is:
\[ \boxed{0.5} \]