Resolu\u00e7\u00e3o<\/b><\/span><\/p>\nConsideremos dois n\u00fameros complexos $$\\begin{array}{*{20}{c}}
\n{{z_A} = a + bi}&{\\text{e}}&{{z_B} = c + di}
\n\\end{array}$$ n\u00e3o nulos, aos quais correspondem os vetores $\\overrightarrow {OA}\u00a0 = \\left( {a,b} \\right)$ e $\\overrightarrow {OB}\u00a0 = \\left( {c,d} \\right)$, respetivamente.<\/p>\n
Determinando o quociente desses n\u00fameros complexos, temos:
\n$$\\begin{array}{*{20}{l}}
\nw& = &{\\frac{{{z_A}}}{{{z_B}}}} \\\\
\n{}& = &{\\frac{{a + bi}}{{c + di}} \\times \\frac{{c – di}}{{c – di}}} \\\\
\n{}& = &{\\frac{{\\left( {ac + bd} \\right) + \\left( {bc – ad} \\right)i}}{{{c^2} + {d^2}}}} \\\\
\n{}& = &{\\frac{{ac + bd}}{{{c^2} + {d^2}}} + \\frac{{bc – ad}}{{{c^2} + {d^2}}}i}
\n\\end{array}$$
\n\u00ad<\/p>\n
\n- Se $w = \\frac{{{z_A}}}{{{z_B}}}$ \u00e9 um n\u00famero real, ent\u00e3o $\\operatorname{Im} (w) = 0$, donde $$\\begin{array}{*{20}{l}}
\n{\\frac{{bc – ad}}{{{c^2} + {d^2}}} = 0}& \\Leftrightarrow &{bc – ad = 0} \\\\
\n{}& \\Leftrightarrow &{ad = bc}
\n\\end{array}$$
\nOu seja, os vetores $\\overrightarrow {OA}\u00a0 = \\left( {a,b} \\right)$ e $\\overrightarrow {OB}\u00a0 = \\left( {c,d} \\right)$ possuem coordenadas diretamente proporcionais, pelo que s\u00e3o colineares.
\n\u00ad<\/li>\n - Se $w = \\frac{{{z_A}}}{{{z_B}}}$ \u00e9 um n\u00famero imagin\u00e1rio puro, ent\u00e3o $\\operatorname{Re} (w) = 0$, donde $$\\begin{array}{*{20}{l}}
\n{\\frac{{ac + bd}}{{{c^2} + {d^2}}} = 0}& \\Leftrightarrow &{ac + bd = 0} \\\\
\n{}& \\Leftrightarrow &{\\overrightarrow {OA} .\\overrightarrow {OB}\u00a0 = 0}
\n\\end{array}$$
\nOu seja, os vetores $\\overrightarrow {OA}\u00a0 = \\left( {a,b} \\right)$ e $\\overrightarrow {OB}\u00a0 = \\left( {c,d} \\right)$ s\u00e3o perpendiculares.
\n\u00ad<\/li>\n<\/ol>\n