If a, b, c are sides of a scalene triangle, t | vedclass

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Question

$\left| {\begin{array}{*{20}{c}}
a&b&c\
b&c&a\
c&a&b
\end{array}} ight| = \left| {\begin{array}{*{20}{c}}
{a + b + c}

Vedclass · Accepted Answer

negative

Vedclass · Answer

$\left| {\begin{array}{*{20}{c}}
a&b&c\
b&c&a\
c&a&b
\end{array}} ight| = \left| {\begin{array}{*{20}{c}}
{a + b + c}&{a + b + c}&{a + b + c}\
b&c&a\
c&a&b
\end{array}} ight|$

             $ = \left( {a + b + c} ight)\begin{array}{*{20}{c}}
1&1&1\
b&c&a\
c&a&b
\end{array}$

             $ = \left( {a + b + c} ight)\begin{array}{*{20}{c}}
0&0&0\
{b - c}&{c - a}&a\
{c - a}&{a - b}&b
\end{array}$

$ = \left( {a + b + c} ight)\left[ {ab + bc + ca - {a^2} - {b^2} - {c^2}} ight]$

$ =  - \left( {a + b + c} ight)\left[ {{{\left( {a - b} ight)}^2} + {{\left( {b - c} ight)}^2} + {{\left( {c - a} ight)}^2}} ight]$

Since $a,b,c$  are sides of $a$ scalene triangle, therfore at least two of the $a,b,c,$ will be unqual.

$	herefore {\left( {a - b} ight)^2} + {\left( {b - c} ight)^2} + {\left( {c - a} ight)^2} > 0$ 

Also $a + b + c > 0$

$	herefore  - \left( {a + b + c} ight)\left[ {{{\left( {a - b} ight)}^2} + {{\left( {b - c} ight)}^2} + {{\left( {c - a} ight)}^2}} ight] < 0$

If $a, b, c$ are sides of a scalene triangle, then the value of $\left| \begin{array}{*{20}{c}}
a&b&c\\
b&c&a\\
c&a&b
\end{array} \right|$ is

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