Let r be a vector perpendicular to a + b + c .If r=l(b×c) + m(c×a) + n(a×b) then l+m+n is

r·(a+b+c)=0 ⇒ r ·a+r·b+r·c=0 r·a=l [a b c]r·b=m [a b c]r·c=n [a b c] r·a+r·b+r·c=(l+m+n) [a b c]l+m+n=0

Let r be a vector perpendicular to a + b + c .If r=l(b×c) + m(c×a) + n(a×b) then l+m+n is

r·(a+b+c)=0 ⇒ r ·a+r·b+r·c=0 r·a=l [a b c]r·b=m [a b c]r·c=n [a b c] r·a+r·b+r·c=(l+m+n) [a b c]l+m+n=0

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Let $r$ be a vector perpendicular to $a + b + c .$
If $r = l (b \times c) + m (c \times a) + n (a \times b)$ then $l + m + n$ is

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$3 \sqrt{3}$

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Detailed Solution

$r \cdot (a + b + c) = 0 \Rightarrow r \cdot a + r \cdot b + r \cdot c = 0 \begin{matrix} r \cdot a = l [a b c] \\ r \cdot b = m [a b c] \\ r \cdot c = n [a b c] \end{matrix} \begin{matrix} r \cdot a + r \cdot b + r \cdot c \\ = (l + m + n) [a b c] \\ l + m + n = 0 \end{matrix}$