- #36

#### neilparker62

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Area of Triangle ABC:

$$=a_j(a_i-b_i)-\frac{1}{2} [a_ia_j-b_ib_j +(a_j-b_j)(a_i-b_i)]$$ $$ =a_ja_i-a_jb_i -\frac{1}{2}[a_ia_j-b_ib_j+a_ja_i-a_jb_i-b_ja_i+b_jb_i]$$ $$=\frac{1}{2}(b_ja_i-b_ia_j)$$ Or: $$=\frac{1}{2}|a||b| \sin(\theta_{B}-\theta_{A})\\=\frac{1}{2}|a||b| (\sin\theta_{B}\cos\theta_{A}-\cos\theta_{B}\sin\theta_{A})$$ $$=\frac{1}{2}|a||b|\left( \frac{b_{j} a_{i} }{|b||a|} - \frac{b_{i} a_{j} }{|b||a|}\right)$$ $$=\frac{1}{2}(b_{j}a_{i}-b_{i}a{j}) $$ All roads lead to Rome!