Question Number 190811 by otchereabdullai last updated on 12/Apr/23
$$\:{If}\:{the}\:{angle}\:{between}\:{the}\:{vectors} \\ $$$$\:{c}\:={ai}+\mathrm{2}{j}\:{and}\:\:{d}=\mathrm{3}{i}+{j}\:{is}\:\mathrm{45}°\:,\:{find} \\ $$$$\:{the}\:{two}\:{possible}\:{values}\:{of}\:{a} \\ $$
Answered by safojontoshtemirov last updated on 12/Apr/23
$${c}=\left({a};\mathrm{2}\right)\:{d}=\left(\mathrm{3};\mathrm{1}\right) \\ $$$${cd}=\mid{c}\mid\mid{d}\mid{cosa} \\ $$$$\mathrm{3}{a}+\mathrm{2}=\sqrt{{a}^{\mathrm{2}} +\mathrm{4}}\centerdot\sqrt{\mathrm{5}} \\ $$$$\mathrm{9}{a}^{\mathrm{2}} +\mathrm{12}{a}+\mathrm{4}=\mathrm{5}{a}^{\mathrm{2}} +\mathrm{20} \\ $$$$\mathrm{4}{a}^{\mathrm{2}} +\mathrm{12}{a}−\mathrm{16}=\mathrm{0} \\ $$$${a}^{\mathrm{2}} +\mathrm{3}{a}−\mathrm{4}=\mathrm{0} \\ $$$${a}=−\mathrm{4}\:\:{a}=\mathrm{1} \\ $$
Commented by otchereabdullai last updated on 12/Apr/23
$${God}\:{bless}\:{you}\:{sir}! \\ $$