Question

If \( a, b, c \) are \( p^{th}, q^{th} \) and \( r^{th} \) terms of a G.P, then the vectors \( \log a \,\hat{i} + \log b \,\hat{j} + \log c \,\hat{k} \) and \( (q-r)\hat{i} + (r-p)\hat{j} + (p-q)\hat{k} \) are

• A. equal
• B. parallel
• C. perpendicular
• D. None of these

Hint:

For G.P. problems involving logs, convert powers into linear form using logarithm properties.

Answer 1

The Correct Option is C

Concept: Vectors are perpendicular if their dot product is zero.

Step 1: Express terms of G.P. Let G.P. be \( A, AR, AR^2, \dots \) \[ a = AR^{p-1}, \quad b = AR^{q-1}, \quad c = AR^{r-1} \] Taking log: \[ \log a = \log A + (p-1)\log R \] Similarly for \( b, c \)

Step 2: Form first vector: \[ \vec{V_1} = (\log A + (p-1)\log R)\hat{i} + (\log A + (q-1)\log R)\hat{j} + (\log A + (r-1)\log R)\hat{k} \]

Step 3: Dot product with second vector: \[ \vec{V_1} \cdot \vec{V_2} \] Terms involving \( \log A \) cancel: \[ (q-r) + (r-p) + (p-q) = 0 \] Remaining terms also cancel $\Rightarrow$ Dot product = 0 Final Answer: \[ \text{Vectors are perpendicular} \]