Mathematics - Vector Question with Solution | TestHub

Let position vector of$P\left(\overrightarrow{p}\right), Q\left(\overrightarrow{q}\right), R\left(\overrightarrow{r}\right) \mathrm{a}\mathrm{n}\mathrm{d} S\left(\overrightarrow{r}\right)$with respect to$O\left(\overrightarrow{o}\right)$Now,$\overrightarrow{OP} . \overrightarrow{OQ}+ \overrightarrow{OR} . \overrightarrow{OS}= \overrightarrow{OR} . \overrightarrow{OP}+ \overrightarrow{OQ} . \overrightarrow{OS}$$\Rightarrow \overrightarrow{p}.\overrightarrow{q}+ \overrightarrow{r}.\overrightarrow{s}= \overrightarrow{r}.\overrightarrow{p}+ \overrightarrow{q}. \overrightarrow{s}$$\Rightarrow \left(\overrightarrow{p}- \overrightarrow{s}\right) . \left(\overrightarrow{q}- \overrightarrow{r}\right)=0$.....(i) Also,$\overrightarrow{OR} . \overrightarrow{OP}+ \overrightarrow{OQ} . \overrightarrow{OS}= \overrightarrow{OQ}. \overrightarrow{OR}+ \overrightarrow{OP} . \overrightarrow{OS}$$\Rightarrow \overrightarrow{r}.\overrightarrow{p}+ \overrightarrow{q} . \overrightarrow{s}= \overrightarrow{q} . \overrightarrow{r}+ \overrightarrow{p} . \overrightarrow{s}$$\Rightarrow \left(\overrightarrow{r}- \overrightarrow{s}\right) . \left(\overrightarrow{p}- \overrightarrow{q}\right)=0$.....(ii) Also$\overrightarrow{OP} . \overrightarrow{OQ}+ \overrightarrow{OR} . \overrightarrow{OS}= \overrightarrow{OQ} . \overrightarrow{OR}+ \overrightarrow{OP} . \overrightarrow{OS}$$\Rightarrow \overrightarrow{p}.\overrightarrow{q}+ \overrightarrow{r}. \overrightarrow{s}= \overrightarrow{q} . \overrightarrow{r}+ \overrightarrow{p}. \overrightarrow{s}$$\Rightarrow \left(\overrightarrow{q}- \overrightarrow{s}\right) . \left(\overrightarrow{p}- \overrightarrow{r}\right)=0$.....(iii)$\Rightarrow$Triangle PQR has S as its orthocenter$\therefore$option (Orthocentre) is correct.