Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}|=1, |\vec{b}|=4$ and $\vec{a} \cdot \vec{b}=2$. If $\vec{c}=(2\vec{a} \times \vec{b})-3\vec{b}$ and the angle between $\vec{b}$ and $\vec{c}$ is $\alpha$, then $192\sin^{2}\alpha$ is equal to ____

JEE Main · 2024 · 31 Jan 2024 · Shift 1 · Vector Algebra

Let

a

and

b

be two vectors such that

∣ a ∣ = 1, ∣ b ∣ = 4

and

a \cdot b = 2

. If

c = (2 a \times b) - 3 b

and the angle between

b

and

c

α

, then

192 sin^{2} α

is equal to ____

Answer: 48

c = 2 (a \times b) - 3 b

. Since

a \times b ⊥ b

b \cdot c = 2 b \cdot (a \times b) - 3∣ b ∣^{2} = 0 - 3 (16) = - 48

∣ a \times b ∣^{2} = ∣ a ∣^{2} ∣ b ∣^{2} - (a \cdot b)^{2} = (1) (16) - 4 = 12

∣ c ∣^{2} = ∣2 (a \times b) - 3 b ∣^{2} = 4∣ a \times b ∣^{2} - 12 (a \times b) \cdot b + 9∣ b ∣^{2} = 4 (12) - 0 + 9 (16) = 48 + 144 = 192

cos α = \frac{b \cdot c}{∣ b ∣∣ c ∣} = \frac{- 48}{4 192} = \frac{- 48}{4 \cdot 8 3} = - \frac{3}{2}

sin^{2} α = 1 - \frac{3}{4} = \frac{1}{4} ⟹ 192 sin^{2} α = 192 \times \frac{1}{4} = 48

JEE Main · 2024 · 31 Jan 2024 · Shift 1 · Integer answer