If $\mathrm{P}, \mathrm{Q}, \mathrm{R}$ are three points such that $|\overrightarrow{P Q}|=3 \&|\ove

Question

If $\mathrm{P}, \mathrm{Q}, \mathrm{R}$ are three points such that $|\overrightarrow{P Q}|=3 \&|\overrightarrow{Q R}|=4$, then the value of $\overrightarrow{P R} \cdot(\overrightarrow{Q R}+\overrightarrow{Q P})$ is

Professor PC Thomas Classes · Answer

Let $|(\vec{q}-\vec{p})|=3 \Rightarrow \vec{q}^2+\vec{p}^2-2 \vec{p} \cdot \vec{q}=9 \ldots$
$
|\vec{r}-\vec{q}|=4 \Rightarrow \vec{q}^{-2}+\vec{r}^2-2 \vec{r} \cdot \vec{q}=16
$
$
\begin{aligned}
& (2)-(1) \Rightarrow\left(\vec{r}^2-\vec{p}^2ight)=2 \vec{q}(\vec{r}-\vec{p})=7 \
& (\vec{r}-\vec{p}) \cdot(\vec{r}+\vec{p}-2 \vec{q})=7 \
& \overrightarrow{P R}(\overrightarrow{Q R}+\overrightarrow{Q p})=7
\end{aligned}
$