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 ⃗ \overrightarrow{\mathrm{c}}=2(\vec{a} \times \vec{b})-3 \vec{b} c = 2 ( a × b ) − 3 b Dot product with a ⃗ \vec{a} a c → . a → = − 6 \overrightarrow{\mathrm{c}} . \overrightarrow{\mathrm{a}}=-6 c . a = − 6
Dot product with b ⃗ \vec{b} b b ⃗ . c → = − 48 \vec{b} . \overrightarrow{\mathrm{c}}=-48 b . c = − 48
c → . c → = 4 ∣ a → × b → ∣ 2 + 9 ∣ b → ∣ 2 \overrightarrow{\mathrm{c}} . \overrightarrow{\mathrm{c}}=4|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|^{2}+9|\overrightarrow{\mathrm{~b}}|^{2} c . c = 4∣ a × b ∣ 2 + 9∣ b ∣ 2
∣ c → ∣ 2 = 4 [ ∣ a ∣ 2 ∣ b ∣ 2 − ( a ⋅ b ⃗ ) 2 ] + 9 ∣ b ⃗ ∣ 2 |\overrightarrow{\mathrm{c}}|^{2}=4\left[|a|^{2}|b|^{2}-(a \cdot \vec{b})^{2}\right]+9|\vec{b}|^{2} ∣ c ∣ 2 = 4 [ ∣ a ∣ 2 ∣ b ∣ 2 − ( a ⋅ b ) 2 ] + 9∣ b ∣ 2 ∣ c → ∣ 2 = 4 [ ( 1 ) ( 4 ) 2 − ( 4 ) ] + 9 ( 16 ) |\overrightarrow{\mathrm{c}}|^{2}=4\left[(1)(4)^{2}-(4)\right]+9(16) ∣ c ∣ 2 = 4 [ ( 1 ) ( 4 ) 2 − ( 4 ) ] + 9 ( 16 )
∣ c → ∣ 2 = 4 [ 12 ] + 144 = 48 + 144 = 192 |\overrightarrow{\mathrm{c}}|^{2}=4[12]+144=48+144=192 ∣ c ∣ 2 = 4 [ 12 ] + 144 = 48 + 144 = 192
∴ cos θ = b → ⋅ c → ∣ b → ∣ ∣ c → ∣ \therefore \cos \theta=\frac{\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}}{|\overrightarrow{\mathrm{b}}||\overrightarrow{\mathrm{c}}|} ∴ cos θ = ∣ b ∣∣ c ∣ b ⋅ c
∴ cos θ = − 48 192.4 = − 3 2 3 \therefore \cos \theta=\frac{-48}{\sqrt{192.4}}=\frac{-3}{2 \sqrt{3}} ∴ cos θ = 192.4 − 48 = 2 3 − 3
⇒ θ = cos − 1 ( − 3 2 ) \Rightarrow \theta=\cos ^{-1}\left(\frac{-\sqrt{3}}{2}\right) ⇒ θ = cos − 1 ( 2 − 3 )