Given: ∣ a ⃗ ∣ = ∣ b ⃗ ∣ = ∣ c ⃗ ∣ = ∣ d ⃗ ∣ = 1 \text{Given: } |\vec{a}| = |\vec{b}| = |\vec{c}| = |\vec{d}| = 1 Given: ∣ a ∣ = ∣ b ∣ = ∣ c ∣ = ∣ d ∣ = 1 ( a ⃗ × b ⃗ ) ⋅ ( c ⃗ × d ⃗ ) = 1 (\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) = 1 ( a × b ) ⋅ ( c × d ) = 1 a ⃗ ⋅ c ⃗ = 1 2 \vec{a} \cdot \vec{c} = \dfrac{1}{2} a ⋅ c = 2 1 To find: which statement is correct? \text{To find: which statement is correct?} To find: which statement is correct? ( 1 ) a ⃗ , b ⃗ , c ⃗ non-coplanar ( 2 ) b ⃗ , c ⃗ , d ⃗ non-coplanar (1)\ \vec{a},\vec{b},\vec{c}\ \text{non-coplanar} \qquad (2)\ \vec{b},\vec{c},\vec{d}\ \text{non-coplanar} ( 1 ) a , b , c non-coplanar ( 2 ) b , c , d non-coplanar ( 3 ) b ⃗ , d ⃗ non-parallel ( 4 ) a ⃗ ∥ d ⃗ , b ⃗ ∥ c ⃗ (3)\ \vec{b},\vec{d}\ \text{non-parallel} \qquad (4)\ \vec{a}\parallel\vec{d},\ \vec{b}\parallel\vec{c} ( 3 ) b , d non-parallel ( 4 ) a ∥ d , b ∥ c Plan: 1. ∣ a ⃗ × b ⃗ ∣ , ∣ c ⃗ × d ⃗ ∣ ≤ 1 \text{Plan: } 1.\ |\vec{a} \times \vec{b}|,\ |\vec{c} \times \vec{d}| \le 1 Plan: 1. ∣ a × b ∣ , ∣ c × d ∣ ≤ 1 2. a ⃗ × b ⃗ = c ⃗ × d ⃗ ⇒ a ⃗ , b ⃗ , c ⃗ , d ⃗ coplanar 2.\ \vec{a} \times \vec{b} = \vec{c} \times \vec{d} \Rightarrow \vec{a},\vec{b},\vec{c},\vec{d} \text{ coplanar} 2. a × b = c × d ⇒ a , b , c , d coplanar 3. Check each option 3.\ \text{Check each option} 3. Check each option ∣ a ⃗ × b ⃗ ∣ = ∣ a ⃗ ∣ ∣ b ⃗ ∣ sin θ = sin θ ≤ 1 |\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta = \sin\theta \le 1 ∣ a × b ∣ = ∣ a ∣∣ b ∣ sin θ = sin θ ≤ 1 ∣ c ⃗ × d ⃗ ∣ = sin ϕ ≤ 1 |\vec{c} \times \vec{d}| = \sin\phi \le 1 ∣ c × d ∣ = sin ϕ ≤ 1 ( a ⃗ × b ⃗ ) ⋅ ( c ⃗ × d ⃗ ) = ∣ a ⃗ × b ⃗ ∣ ∣ c ⃗ × d ⃗ ∣ cos α = 1 (\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) = |\vec{a} \times \vec{b}|\,|\vec{c} \times \vec{d}|\cos\alpha = 1 ( a × b ) ⋅ ( c × d ) = ∣ a × b ∣ ∣ c × d ∣ cos α = 1 ⇒ ∣ a ⃗ × b ⃗ ∣ = 1 , ∣ c ⃗ × d ⃗ ∣ = 1 , cos α = 1 \Rightarrow |\vec{a} \times \vec{b}| = 1,\quad |\vec{c} \times \vec{d}| = 1,\quad \cos\alpha = 1 ⇒ ∣ a × b ∣ = 1 , ∣ c × d ∣ = 1 , cos α = 1 a ⃗ × b ⃗ = c ⃗ × d ⃗ = n ^ \vec{a} \times \vec{b} = \vec{c} \times \vec{d} = \hat{n} a × b = c × d = n ^ ∣ a ⃗ × b ⃗ ∣ = 1 ⇒ a ⃗ ⋅ b ⃗ = 0 |\vec{a} \times \vec{b}| = 1 \Rightarrow \vec{a} \cdot \vec{b} = 0 ∣ a × b ∣ = 1 ⇒ a ⋅ b = 0 ∣ c ⃗ × d ⃗ ∣ = 1 ⇒ c ⃗ ⋅ d ⃗ = 0 |\vec{c} \times \vec{d}| = 1 \Rightarrow \vec{c} \cdot \vec{d} = 0 ∣ c × d ∣ = 1 ⇒ c ⋅ d = 0 a ⃗ , b ⃗ ⊥ n ^ and c ⃗ , d ⃗ ⊥ n ^ \vec{a},\vec{b} \perp \hat{n} \quad \text{and} \quad \vec{c},\vec{d} \perp \hat{n} a , b ⊥ n ^ and c , d ⊥ n ^ ∴ a ⃗ , b ⃗ , c ⃗ , d ⃗ lie in the plane ⊥ n ^ \therefore \vec{a},\vec{b},\vec{c},\vec{d} \text{ lie in the plane } \perp \hat{n} ∴ a , b , c , d lie in the plane ⊥ n ^ ∴ a ⃗ , b ⃗ , c ⃗ coplanar and b ⃗ , c ⃗ , d ⃗ coplanar \therefore \vec{a},\vec{b},\vec{c} \text{ coplanar and } \vec{b},\vec{c},\vec{d} \text{ coplanar} ∴ a , b , c coplanar and b , c , d coplanar ⇒ ( 1 ) , ( 2 ) are false \Rightarrow (1),\ (2) \text{ are false} ⇒ ( 1 ) , ( 2 ) are false n ^ × a ⃗ = ( a ⃗ × b ⃗ ) × a ⃗ = b ⃗ \hat{n} \times \vec{a} = (\vec{a} \times \vec{b}) \times \vec{a} = \vec{b} n ^ × a = ( a × b ) × a = b n ^ × c ⃗ = d ⃗ \hat{n} \times \vec{c} = \vec{d} n ^ × c = d If b ⃗ ∥ d ⃗ ⇒ n ^ × a ⃗ ∥ n ^ × c ⃗ \text{If } \vec{b} \parallel \vec{d} \Rightarrow \hat{n} \times \vec{a} \parallel \hat{n} \times \vec{c} If b ∥ d ⇒ n ^ × a ∥ n ^ × c ⇒ a ⃗ ∥ c ⃗ ⇒ a ⃗ ⋅ c ⃗ = ± 1 \Rightarrow \vec{a} \parallel \vec{c} \Rightarrow \vec{a} \cdot \vec{c} = \pm 1 ⇒ a ∥ c ⇒ a ⋅ c = ± 1 But a ⃗ ⋅ c ⃗ = 1 2 ✗ \text{But } \vec{a} \cdot \vec{c} = \dfrac{1}{2} \ \ ✗ But a ⋅ c = 2 1 ✗ ∴ b ⃗ , d ⃗ are non-parallel ⇒ ( 3 ) is true \therefore \vec{b}, \vec{d} \text{ are non-parallel} \Rightarrow (3) \text{ is true} ∴ b , d are non-parallel ⇒ ( 3 ) is true b ⃗ ∥ c ⃗ ⇒ a ⃗ ⋅ c ⃗ = ± a ⃗ ⋅ b ⃗ = 0 \vec{b} \parallel \vec{c} \Rightarrow \vec{a} \cdot \vec{c} = \pm \vec{a} \cdot \vec{b} = 0 b ∥ c ⇒ a ⋅ c = ± a ⋅ b = 0 But a ⃗ ⋅ c ⃗ = 1 2 ⇒ ( 4 ) is false \text{But } \vec{a} \cdot \vec{c} = \dfrac{1}{2} \Rightarrow (4) \text{ is false} But a ⋅ c = 2 1 ⇒ ( 4 ) is false ( 3 ) b ⃗ , d ⃗ are non-parallel \boxed{(3)\ \vec{b}, \vec{d} \text{ are non-parallel}} ( 3 ) b , d are non-parallel
