Question Number 25971 by Tinkutara last updated on 17/Dec/17
$${In}\:{finding}\:{the}\:{equations}\:{of}\:{the} \\ $$$${bisectors}\:{of}\:{the}\:{angles}\:{between}\:{two} \\ $$$${lines}\:{a}_{\mathrm{1}} {x}+{b}_{\mathrm{1}} {y}+{c}_{\mathrm{1}} =\mathrm{0}\:{and}\:{a}_{\mathrm{2}} {x}+{b}_{\mathrm{2}} {y}+{c}_{\mathrm{2}} =\mathrm{0}, \\ $$$${why}\:{we}\:{observe}\:{a}_{\mathrm{1}} {a}_{\mathrm{2}} +{b}_{\mathrm{1}} {b}_{\mathrm{2}} >\mathrm{0}\:{or}\:<\mathrm{0} \\ $$$${for}\:{obtuse}\:{and}\:{acute}\:{angle}\:{bisectors}? \\ $$
Commented by Tinkutara last updated on 17/Dec/17
Commented by Tinkutara last updated on 17/Dec/17
$${For}\:{example}\:{here},\:{why}\:{we}\:{are} \\ $$$${calculating}\:{a}_{\mathrm{1}} {a}_{\mathrm{2}} +{b}_{\mathrm{1}} {b}_{\mathrm{2}} ?\:{Why}\:{not} \\ $$$${a}_{\mathrm{1}} {b}_{\mathrm{1}} +{a}_{\mathrm{2}} {b}_{\mathrm{2}} \:{or}\:{something}\:{else}? \\ $$
Answered by ajfour last updated on 17/Dec/17
$${to}\:{check}\:{if}\:\mid\theta_{\mathrm{2}} −\theta_{\mathrm{1}} \mid\:{is}\:{obtuse}\:{or} \\ $$$${acute}\:\:{we}\:{look}\:{for}\:{sign}\:{of}\: \\ $$$$\mathrm{cos}\:\left(\theta_{\mathrm{2}} −\theta_{\mathrm{1}} \right). \\ $$$$\mid\theta_{\mathrm{2}} −\theta_{\mathrm{1}} \mid\:{is}\:{acute}\:{only}\:{if} \\ $$$$\mathrm{cos}\:\left(\theta_{\mathrm{2}} −\theta_{\mathrm{1}} \right)\:>\:\mathrm{0}\:,\:\:{and} \\ $$$$\mathrm{cos}\:\theta_{\mathrm{1}} =\frac{−{b}_{\mathrm{1}} }{\:\sqrt{{a}_{\mathrm{1}} ^{\mathrm{2}} +{b}_{\mathrm{1}} ^{\mathrm{2}} }}\:\:,\mathrm{sin}\:\theta_{\mathrm{1}} =\frac{{a}_{\mathrm{1}} }{\:\sqrt{{a}_{\mathrm{1}} ^{\mathrm{2}} +{b}_{\mathrm{1}} ^{\mathrm{2}} }} \\ $$$$\:\mathrm{cos}\:\theta_{\mathrm{2}} =\frac{−{b}_{\mathrm{2}} }{\:\sqrt{{a}_{\mathrm{2}} ^{\mathrm{2}} +{b}_{\mathrm{2}} ^{\mathrm{2}} }}\:,\:\mathrm{sin}\:\theta_{\mathrm{2}} =\frac{{a}_{\mathrm{2}} }{\:\sqrt{{a}_{\mathrm{2}} ^{\mathrm{2}} +{b}_{\mathrm{2}} ^{\mathrm{2}} }}\:\:\:\: \\ $$$${so}\:\:\mathrm{cos}\:\left(\theta_{\mathrm{2}} −\theta_{\mathrm{1}} \right)\:>\:\mathrm{0} \\ $$$$\Rightarrow\:\:\mathrm{cos}\:\theta_{\mathrm{2}} \mathrm{cos}\:\theta_{\mathrm{1}} +\mathrm{sin}\:\theta_{\mathrm{2}} \mathrm{sin}\:\theta_{\mathrm{1}} \:>\:\mathrm{0} \\ $$$${or}\:\:\:\:{b}_{\mathrm{1}} {b}_{\mathrm{2}} +{a}_{\mathrm{1}} {a}_{\mathrm{2}} \:>\:\mathrm{0}\:. \\ $$$$\:\:\:\:\:\: \\ $$
Commented by Tinkutara last updated on 17/Dec/17
$${What}\:{are}\:\theta_{\mathrm{1}} \:{and}\:\theta_{\mathrm{2}} ? \\ $$
Commented by ajfour last updated on 17/Dec/17
$${Angles}\:\:{of}\:{lines}\:{with}\:+{ve}\:{x}\:{axis}, \\ $$$${equations}\:{of}\:{whose}\:{bisectors}\:{we} \\ $$$${seek}. \\ $$
Commented by Tinkutara last updated on 17/Dec/17
Sir can you prove exactly Step 3?