Question:

Step 1: Set up the perpendicularity condition Since $\vec{r}$ is perpendicular to $(\vec{a}+\vec{b}+\vec{c})$, we have: $\vec{r}·(\vec{a}+\vec{b}+\vec{c}) = 0$ Step 2: Substitute the expression for r $[l(\vec{b}\times\vec{c}) + m(\vec{c}\times\vec{a}) + n(\vec{a}\times\vec{b})]·(\vec{a}+\vec{b}+\vec{c}) = 0$ Step 3: Expand the dot product $l(\vec{b}\times\vec{c})·\vec{a} + l(\vec{b}\times\vec{c})·\vec{b} + l(\vec{b}\times\vec{c})·\vec{c}$ $+ m(\vec{c}\times\vec{a})·\vec{a} + m(\vec{c}\times\vec{a})·\vec{b} + m(\vec{c}\times\vec{a})·\vec{c}$ $+ n(\vec{a}\times\vec{b})·\vec{a} + n(\vec{a}\times\vec{b})·\vec{b} + n(\vec{a}\times\vec{b})·\vec{c} = 0$ Step 4: Apply the property that (u×v)·u = 0 and (u×v)·v = 0 The cross product is perpendicular to both vectors involved, so: - $(\vec{b}\times\vec{c})·\vec{b} = 0$ and $(\vec{b}\times\vec{c})·\vec{c} = 0$ - $(\vec{c}\times\vec{a})·\vec{c} = 0$ and $(\vec{c}\times\vec{a})·\vec{a} = 0$ - $(\vec{a}\times\vec{b})·\vec{a} = 0$ and $(\vec{a}\times\vec{b})·\vec{b} = 0$ Step 5: Simplify using scalar triple product The remaining non-zero terms are: $l[(\vec{b}\times\vec{c})·\vec{a}] + m[(\vec{c}\times\vec{a})·\vec{b}] + n[(\vec{a}\times\vec{b})·\vec{c}] = 0$ Step 6: Use the scalar triple product definition Recall that $(\vec{b}\times\vec{c})·\vec{a} = [\vec{a}\vec{b}\vec{c}]$, $(\vec{c}\times\vec{a})·\vec{b} = [\vec{a}\vec{b}\vec{c}]$, and $(\vec{a}\times\vec{b})·\vec{c} = [\vec{a}\vec{b}\vec{c}]$ All three scalar triple products equal the same value. Given $[\vec{a}\vec{b}\vec{c}] = 2$: Step 7: Substitute and solve $l(2) + m(2) + n(2) = 0$ $2(l + m + n) = 0$ $l + m + n = 0$ ∴ Answer: 0