Q113 / 200 · Manipal (MET) 2014 · Mathematics
If $a, b, c$ are $p^{t h}$ and $q^{\text {th }}$ and $r^{t h}$ terms of a GP, then the vectors $\log a \hat{\mathbf{i}}+\log b \hat{\mathbf{j}}+\log c \hat{\mathbf{k}}$ and $(q-r) \hat{\mathbf{i}}+(r-p) \hat{\mathbf{j}}+(p-q) \hat{\mathbf{k}}$ are
- Aequal
- Bparallel
- Correct answerCperpendicular
- DNone of these
Solution
Let the first term and common ratio of a GP be $\alpha$ and $\beta$, then
$\begin{aligned}
a= & \alpha \cdot \beta^{p-1}, b=\alpha \cdot \beta^{q-1} \text { and } c=\alpha \cdot \beta^{r-1} \\
\therefore \quad & \log a=\log \alpha+(p-1) \log \beta, \\
& \log b=\log \alpha+(q-1) \log \beta \text { and } \\
& \log c=\log \alpha+(r-1) \log \beta
\end{aligned}$
The dot product of the given two vectors is $\begin{aligned} & (q-r) \log a+(r-p) \log b+(p-q) \log c \\ & \Rightarrow(q-r)[\log \alpha+(p-1) \log \beta]+(r-p) \\ & {[\log \alpha+(q-1) \log \beta]+(p-q)[\log \alpha+(r-1) \log \beta]} \\ & \Rightarrow \log \alpha[q-r+r-p+p-q]+ \\ & \log \beta[(p-1)(q-r)+(r-p)(q-1) \\ & \quad+(r-1)(p-q)] \\ & \quad=0+0=0 \end{aligned}$ $\Rightarrow$ The two vectors are perpedicular.
The dot product of the given two vectors is $\begin{aligned} & (q-r) \log a+(r-p) \log b+(p-q) \log c \\ & \Rightarrow(q-r)[\log \alpha+(p-1) \log \beta]+(r-p) \\ & {[\log \alpha+(q-1) \log \beta]+(p-q)[\log \alpha+(r-1) \log \beta]} \\ & \Rightarrow \log \alpha[q-r+r-p+p-q]+ \\ & \log \beta[(p-1)(q-r)+(r-p)(q-1) \\ & \quad+(r-1)(p-q)] \\ & \quad=0+0=0 \end{aligned}$ $\Rightarrow$ The two vectors are perpedicular.
Hence, the correct answer is option: 'C'.