Resumo - Vetores

PROJEÇÃO DE VETORES\n1) Decompor \\( \\mathbf{u} = \\mathbf{u}_{//} + \\mathbf{u}_{\\perp} \\)\n\\[ \\mathbf{u}_{//} \\]\n\\[ \\text{Proj}\\_{\\mathbf{u}} = \\mathbf{u}_{S} = \\mathbf{u}^T \\frac{(\\mathbf{u}\\cdot \\mathbf{v})}{(|\\mathbf{u}|^2)}\\] PRODUTO ESCALAR\n\\[ \\mathbf{u} \\cdot \\mathbf{v} = x_1x_4 + y_1y_4 + z_1z_4 \\]\n\\[ \\mathbf{u} \\perp \\mathbf{v} \\Leftrightarrow \\mathbf{u} \\cdot \\mathbf{v} = 0 \\]\n\\[ \\mathbf{u} \\cdot \\mathbf{v} = |\\mathbf{u}| |\\mathbf{v}| \\cos \\theta \\]\n\\[ \\cos \\theta = \\frac{\\mathbf{u} \\cdot \\mathbf{v}}{|\\mathbf{u}| |\\mathbf{v}|} \\]\n\nPRODUTO VETORIAL\n\\[ \\mathbf{u} \\times \\mathbf{v} = \\left| \\begin{array}{ccc}\n\\mathbf{i} & \\mathbf{j} & \\mathbf{k} \\\n x_1 & y_1 & z_1 \\\n x_2 & y_2 & z_2\\end{array} \\right|\\]\n\\[ \\mathbf{u} \\times \\mathbf{u} = \\mathbf{0} \\]\n\\[ \\mathbf{u} \\times \\mathbf{v} = -\\mathbf{v} \\times \\mathbf{u} \\]\n\\[ |\\mathbf{u} \\times \\mathbf{v}| = |\\mathbf{u}||\\mathbf{v}| \\sin \\theta \\]\n\\[ \\text{sen } \\theta = \\frac{|\\mathbf{u} \\times \\mathbf{v}|}{|\\mathbf{u}| |\\mathbf{v}|} \\]\n\nPRODUTO MISTO\n\\[ [\\mathbf{u}, \\mathbf{v}, \\mathbf{w}] = \\mathbf{u} \\cdot (\\mathbf{v} \\times \\mathbf{w})\\]\n\\[ [\\mathbf{u}, \\mathbf{v}, \\mathbf{w}] = \\left| \\begin{array}{ccc}\n\\mathbf{u} \\ \\mathbf{v} \\ \\mathbf{w}\\end{array} \\right|\\]\n\\[ [\\mathbf{u}, \\mathbf{v}, \\mathbf{w}] = 0 \\Leftrightarrow \\mathbf{u}, \\mathbf{v}, \\mathbf{w} \\text{ coplanares} \\]\n\\[ V_{par = [\\mathbf{u}, \\mathbf{v}, \\mathbf{w}]} = \\text{VOLUME DO PARALELEPÍPEDO}\\]\n\\[ \\text{Vetereádio =} \\frac{[\\mathbf{u}, \\mathbf{v}, \\mathbf{w}]}{3} \\]