Given ∣ a ⃗ ∣ = 1 , ∣ b ⃗ ∣ = 4 , a ⃗ ⋅ b ⃗ = 2 |\vec{a}|=1,|\vec{b}|=4, \vec{a} \cdot \vec{b}=2 ∣ a ∣ = 1 , ∣ b ∣ = 4 , a ⋅ b = 2
c ⃗ = 2 ( a ⃗ × b ⃗ ) − 3 b ⃗ \vec{\mathrm{c}}=2(\vec{\mathrm{a}} \times \vec{\mathrm{b}})-3 \vec{\mathrm{b}} c = 2 ( a × b ) − 3 b
Dot product with a → \overrightarrow{\mathrm{a}} a on both sides
c → . a → = − 6 \overrightarrow{\mathrm{c}} . \overrightarrow{\mathrm{a}}=-6 c . a = − 6 ..... (1)
Dot product with b ⃗ \vec{b} b on both sides
a m p ; b → c → = − 48 ... (2) a m p ; c → ⋅ c → = 4 ∣ a → × b → ∣ 2 + 9 ∣ b → ∣ 2 a m p ; ∣ c → ∣ 2 = 4 [ ∣ a ∣ 2 ∣ b ∣ 2 − ( a ⋅ b → ) 2 ] + 9 ∣ b → ∣ 2 a m p ; ∣ c → ∣ 2 = 4 [ ( 1 ) ( 4 ) 2 − ( 4 ) ] + 9 ( 16 ) a m p ; ∣ c → ∣ 2 = 4 [ 12 ] + 144 a m p ; ∣ c → ∣ 2 = 48 + 144 a m p ; ∣ c → ∣ 2 = 192 a m p ; ∴ cos θ = b → ⋅ c → ∣ b → ∣ c → ∣ a m p ; ∴ cos θ = − 48 192 ⋅ 4 a m p ; ∴ cos θ = − 48 8 3 .4 a m p ; ∴ cos θ = − 3 2 3 a m p ; ∴ cos θ = − 3 2 ⇒ θ = cos − 1 ( − 3 2 ) \begin{aligned}& \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}=-48 \quad \text{... (2)}\\& \overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{c}}=4|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|^2+9|\overrightarrow{\mathrm{b}}|^2 \\& |\overrightarrow{\mathrm{c}}|^2=4\left[|\mathrm{a}|^2|\mathrm{~b}|^2-(\mathrm{a} \cdot \overrightarrow{\mathrm{b}})^2\right]+9|\overrightarrow{\mathrm{b}}|^2 \\& |\overrightarrow{\mathrm{c}}|^2=4\left[(1)(4)^2-(4)\right]+9(16) \\& |\overrightarrow{\mathrm{c}}|^2=4[12]+144 \\& |\overrightarrow{\mathrm{c}}|^2=48+144 \\& |\overrightarrow{\mathrm{c}}|^2=192 \\& \therefore \cos \theta=\frac{\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}}{|\overrightarrow{\mathrm{b}}| \overrightarrow{\mathrm{c}} \mid} \\& \therefore \cos \theta=\frac{-48}{\sqrt{192} \cdot 4} \\& \therefore \cos \theta=\frac{-48}{8 \sqrt{3} .4} \\& \therefore \cos \theta=\frac{-3}{2 \sqrt{3}} \\& \therefore \cos \theta=\frac{-\sqrt{3}}{2} \Rightarrow \theta=\cos ^{-1}\left(\frac{-\sqrt{3}}{2}\right)\end{aligned} am p ; b c = − 48 ... (2) am p ; c ⋅ c = 4∣ a × b ∣ 2 + 9∣ b ∣ 2 am p ; ∣ c ∣ 2 = 4 [ ∣ a ∣ 2 ∣ b ∣ 2 − ( a ⋅ b ) 2 ] + 9∣ b ∣ 2 am p ; ∣ c ∣ 2 = 4 [ ( 1 ) ( 4 ) 2 − ( 4 ) ] + 9 ( 16 ) am p ; ∣ c ∣ 2 = 4 [ 12 ] + 144 am p ; ∣ c ∣ 2 = 48 + 144 am p ; ∣ c ∣ 2 = 192 am p ; ∴ cos θ = ∣ b ∣ c ∣ b ⋅ c am p ; ∴ cos θ = 192 ⋅ 4 − 48 am p ; ∴ cos θ = 8 3 .4 − 48 am p ; ∴ cos θ = 2 3 − 3 am p ; ∴ cos θ = 2 − 3 ⇒ θ = cos − 1 ( 2 − 3 )