a m p ; a → = i ^ + j ^ + k ^ a m p ; b → = 2 i ^ + 2 j ^ + k ^ a m p ; d → = a → × b → a m p ; = − i ^ + j ^ a m p ; ∣ c → − 2 a → ∣ 2 = 8 a m p ; ∣ c ∣ 2 + 4 ∣ a ∣ 2 − 4 ( a ⋅ c ) = 8 a m p ; c 2 + 12 − 4 c = 8 a m p ; c 2 − 4 c + 4 = 0 a m p ; ∣ c ∣ = 2 a m p ; d → = a → × b → a m p ; d → × c → = ( a → × b → ) 2 × c → a m p ; ( ∣ d ∣ ∣ c ∣ sin π 4 ) 2 = ( ( a ⋅ c ) ⋅ b − ( b ⋅ c ) ⋅ a ) 2 a m p ; 4 = 4 b 2 + ( b ⋅ c ) 2 2 ( a ) 2 − 2 ( b ⋅ c ) ( a ⋅ b ) \begin{aligned}& \overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}} \\& \overrightarrow{\mathrm{~b}}=2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}} \\& \overrightarrow{\mathrm{~d}}=\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}} \\& =-\hat{\mathrm{i}}+\hat{\mathrm{j}} \\& |\overrightarrow{\mathrm{c}}-2 \overrightarrow{\mathrm{a}}|^2=8 \\& |\mathrm{c}|^2+4|\mathrm{a}|^2-4(\mathrm{a} \cdot \mathrm{c})=8 \\& \mathrm{c}^2+12-4 \mathrm{c}=8 \\& \mathrm{c}^2-4 \mathrm{c}+4=0 \\& |\mathrm{c}|=2 \\& \overrightarrow{\mathrm{~d}}=\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}} \\& \overrightarrow{\mathrm{~d}} \times \overrightarrow{\mathrm{c}}=(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}})^2 \times \overrightarrow{\mathrm{c}} \\& \left(|\mathrm{~d}||\mathrm{c}| \sin \frac{\pi}{4}\right)^2=((\mathrm{a} \cdot \mathrm{c}) \cdot \mathrm{b}-(\mathrm{b} \cdot \mathrm{c}) \cdot \mathrm{a})^2 \\& 4=4 \mathrm{~b}^2+(\mathrm{b} \cdot \mathrm{c})^2 2(\mathrm{a})^2-2(\mathrm{~b} \cdot \mathrm{c})(\mathrm{a} \cdot \mathrm{~b})\end{aligned} am p ; a = i ^ + j ^ + k ^ am p ; b = 2 i ^ + 2 j ^ + k ^ am p ; d = a × b am p ; = − i ^ + j ^ am p ; ∣ c − 2 a ∣ 2 = 8 am p ; ∣ c ∣ 2 + 4∣ a ∣ 2 − 4 ( a ⋅ c ) = 8 am p ; c 2 + 12 − 4 c = 8 am p ; c 2 − 4 c + 4 = 0 am p ; ∣ c ∣ = 2 am p ; d = a × b am p ; d × c = ( a × b ) 2 × c am p ; ( ∣ d ∣∣ c ∣ sin 4 π ) 2 = (( a ⋅ c ) ⋅ b − ( b ⋅ c ) ⋅ a ) 2 am p ; 4 = 4 b 2 + ( b ⋅ c ) 2 2 ( a ) 2 − 2 ( b ⋅ c ) ( a ⋅ b )
a m p ; Let b . c = x a m p ; a m p ; 4 = 36 + 3 x 2 − 20 x a m p ; 3 x 2 − 20 x + 32 = 0 a m p ; 3 x 2 − 12 x − 8 x + 32 = 0 a m p ; x = 8 3 , 4 a m p ; b.c = 8 3 , 4 a m p ; b.c = 8 3 a m p ; Now ∣ 10 − 3 b . c ∣ + ∣ d × c ∣ 2 a m p ; ∣ 10 − 8 ∣ + ( 2 ) 2 a m p ; ⇒ 6 Ans. \begin{aligned}&\text { Let } \mathrm{b} . \mathrm{c}=\mathrm{x}\\&\begin{aligned}& 4=36+3 x^2-20 x \\& 3 x^2-20 x+32=0 \\& 3 x^2-12 x-8 x+32=0 \\& x=\frac{8}{3}, 4 \\& \text { b.c }=\frac{8}{3}, 4 \\& \text { b.c }=\frac{8}{3} \\& \text { Now }|10-3 \mathrm{~b} . \mathrm{c}|+|\mathrm{d} \times \mathrm{c}|^2 \\& |10-8|+(2)^2 \\& \Rightarrow 6 \text { Ans. }\end{aligned}\end{aligned} am p ; Let b . c = x am p ; am p ; 4 = 36 + 3 x 2 − 20 x am p ; 3 x 2 − 20 x + 32 = 0 am p ; 3 x 2 − 12 x − 8 x + 32 = 0 am p ; x = 3 8 , 4 am p ; b.c = 3 8 , 4 am p ; b.c = 3 8 am p ; Now ∣10 − 3 b . c ∣ + ∣ d × c ∣ 2 am p ; ∣10 − 8∣ + ( 2 ) 2 am p ; ⇒ 6 Ans.