a m p ; a → = ( 2 , 1 , − 3 ) a m p ; b → = ( − 1 , − 3 , − 5 ) a m p ; p → × q → = ∣ i ^ a m p ; j ^ a m p ; k ^ 1 a m p ; 2 a m p ; − 3 2 a m p ; 4 a m p ; − 5 ∣ a m p ; = 2 i ^ − j ^ a m p ; b → − a → = − 3 i ^ − 4 j ^ − 2 k ^ a m p ; S d = ∣ ( b → − a → ) ⋅ ( p → × q → ) ∣ ∣ p → × q → ∣ a m p ; = 2 5 a m p ; ( S d ) 2 = 4 5 a m p ; m = 4 , n = 5 ⇒ m + n = 9 \begin{aligned}& \overrightarrow{\mathrm{a}}=(2,1,-3) \\& \overrightarrow{\mathrm{b}}=(-1,-3,-5) \\& \overrightarrow{\mathrm{p}} \times \overrightarrow{\mathrm{q}}=\left|\begin{array}{ccc}\hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\1 & 2 & -3 \\2 & 4 & -5\end{array}\right| \\& =2 \hat{\mathrm{i}}-\hat{\mathrm{j}} \\& \overrightarrow{\mathrm{~b}}-\overrightarrow{\mathrm{a}}=-3 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}-2 \hat{\mathrm{k}} \\& \mathrm{~S}_{\mathrm{d}}=\frac{|(\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}}) \cdot(\overrightarrow{\mathrm{p}} \times \overrightarrow{\mathrm{q}})|}{|\overrightarrow{\mathrm{p}} \times \overrightarrow{\mathrm{q}}|} \\& =\frac{2}{\sqrt{5}} \\& \left(\mathrm{~S}_{\mathrm{d}}\right)^2=\frac{4}{5} \\& \mathrm{~m}=4, \mathrm{n}=5 \Rightarrow \mathrm{~m}+\mathrm{n}=9\end{aligned} am p ; a = ( 2 , 1 , − 3 ) am p ; b = ( − 1 , − 3 , − 5 ) am p ; p × q = i ^ 1 2 am p ; j ^ am p ; 2 am p ; 4 am p ; k ^ am p ; − 3 am p ; − 5 am p ; = 2 i ^ − j ^ am p ; b − a = − 3 i ^ − 4 j ^ − 2 k ^ am p ; S d = ∣ p × q ∣ ∣ ( b − a ) ⋅ ( p × q ) ∣ am p ; = 5 2 am p ; ( S d ) 2 = 5 4 am p ; m = 4 , n = 5 ⇒ m + n = 9