Question

In a right angled $\Delta ABC$. the hypotenuse $AB=p$, then $\overrightarrow{AB}.\overrightarrow{AC}+\overrightarrow{BC}.\overrightarrow{BA}+\overrightarrow{CA}.\overrightarrow{CB}$ is

• A. $2p^2$
• B. $\frac{p^2}{2}$
• C. $p^2$
• D. $0$

Answer 1

Answer: (C)

$p^2$

Let P.V of $A,B,C$ be $\overline{a},\overline{b},\overline{o}$ respectively.
$=(\overline{b}-\overline{a})\cdot (-\overline{a})+(-\overline{b})\cdot (\overline{a}-\overline{b})+\left(\overline{a}\right)\cdot \overline{b}$
$=-\overline{a}\cdot \overline{b}+|\overline{a}{|}^2-\overline{a}\cdot \overline{b}+|\overline{b}{|}^2+\overline{a}\cdot \overline{b}$
$=|\overline{a}{|}^2+|\overline{b}{|}^2$ $(\overline{a}\cdot \overline{b}=0)$
$=|AB{|}^2$
$=p^2$