Question

If $\left|\begin{array}{c}\overline{a}\end{array}\right|=5,\left|\begin{array}{c}\overline{b}\end{array}\right|=4,\left|\begin{array}{c}\overline{c}\end{array}\right|=3$, then the value of $\left|\begin{array}{c}\overline{a}.\overline{b}+\overline{b}.\overline{c}+\overline{c}.\overline{a}\end{array}\right|$ equals, if $\overline{a}+\overline{b}+\overline{c}=0$

• A. $25$
• B. $50$
• C. $-25$
• D. $-50$

Answer 1

Answer: (A)

$25$

Consider $\overline{a}+\overline{b}+\overline{c}=0$

$\Rightarrow |\overline{a}{|}^2+|\overline{b}{|}^2+|\overline{c}{|}^2+2[\overline{a}\cdot \overline{b}+\overline{b}\cdot \overline{c}+\overline{c}\cdot \overline{a}]=0$
$\Rightarrow \overline{a}\cdot \overline{b}+\overline{b}\cdot \overline{c}+\overline{c}\cdot \overline{a}=\frac{-1}{2}[{\overline{a}}^2+{\overline{b}}^2+{\overline{c}}^2]$
$\Rightarrow |\overline{a}\cdot \overline{b}+\overline{b}\cdot \overline{c}+\overline{c}\cdot \overline{a}|=\left|\frac{-1}{2}\right(50\left)\right|=25$