A) \[\frac{\pi }{6}\]
B) \[\frac{\pi }{4}\]
C) \[\frac{\pi }{3}\]
D) \[\frac{\pi }{2}\]
Correct Answer: D
Solution :
| [d] Let A be the first and x the common ration of G.P. |
| So, \[a=A{{x}^{p-1}}\Rightarrow \log a=\log A+(p-1)\log \,x\] |
| Similarly, \[\log \,b=\log A+(q-1)log\,x\] |
| and \[\log \,c=\log A+(r-1)log\,x\] |
| if \[\overset{\to }{\mathop{\alpha }}\,=\log \,{{a}^{2}}\hat{i}+\log {{b}^{2}}\hat{j}+\log \,{{c}^{2}}\hat{k}\] |
| and \[\overset{\to }{\mathop{\beta }}\,=(q-r)\hat{i}+(r-p)\hat{j}+(p-q)\hat{k}\] then |
| \[\overset{\to }{\mathop{\alpha }}\,.\overset{\to }{\mathop{\beta }}\,=2[log\,a(q-r)+log\,b(r-p)+log\,c(p-q)]\] |
| \[=2[(q-r)\{log\,A+(p-1)log\,x\}\] |
| \[+(r-p)\{log\,A+(q-1)log\,x\}\] |
| \[+(p-q)\{log\,A+(r-1)log\,x\}]\] |
| \[=2[(q-r+r-p+p-q)log\,A\] |
| \[+(qp-pr-p+r+qr-pq\] |
| \[-r+p+pr-qr-p+q)\log x]=0\] |
| Hence, the angle between \[\overset{\to }{\mathop{\alpha }}\,\] and \[\overset{\to }{\mathop{\beta }}\,\] is \[\frac{\pi }{2}.\] |
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