Question

If the vectors $a,b,$ and $c$ from the sides $BC,CA,$ and $AB,$ respectively of a $△ABC$'s, then

• A. $a·b+b·c+c·a=0$
• B. $a\times b=b\times c=c\times a$
• C. $a·b=b·c=c·a$
• D. $a\times b+b\times c+c\times a=0$

Answer 1

The correct option is B

$a\times b=b\times c=c\times a$

Explanation for the correct option :

Applying the triangle law ,

As vectors $a,b,c$ form the side of $△ABC$'s so by triangle law $a+b+c=0$.

Take cross product by $a$ on both sides:

$\begin{array}{rcl}a\times \left(a+b+c\right)& =& a\times 0\\ a\times a+a\times b+a\times c& =& 0\\ 0+a\times b+a\times c& =& 0\\ a\times b& =& -(a\times c)\\ a\times b& =& c\times a\end{array}$

Similarly by taking cross product by $b$ and $c$ it can be shown that $a\times b=b\times c$ and $b\times c=c\times a$ respectively.

Thus, it is shown that $a\times b=b\times c=c\times a$.

Hence, the correct option is B.