Δ P Q R : a ⃗ = Q R → , b ⃗ = R P → , c ⃗ = P Q → \Delta PQR:\ \vec{a} = \overrightarrow{QR},\ \vec{b} = \overrightarrow{RP},\ \vec{c} = \overrightarrow{PQ} Δ P QR : a = QR , b = R P , c = P Q ∣ a ⃗ ∣ = 12 , ∣ b ⃗ ∣ = 4 3 , b ⃗ ⋅ c ⃗ = 24 |\vec{a}| = 12,\quad |\vec{b}| = 4\sqrt{3},\quad \vec{b} \cdot \vec{c} = 24 ∣ a ∣ = 12 , ∣ b ∣ = 4 3 , b ⋅ c = 24 To find: which of ( 1 ) , ( 2 ) , ( 3 ) , ( 4 ) are true \text{To find: which of }(1),(2),(3),(4)\text{ are true} To find: which of ( 1 ) , ( 2 ) , ( 3 ) , ( 4 ) are true a ⃗ = Q R → , b ⃗ = R P → , c ⃗ = P Q → \vec{a} = \overrightarrow{QR}, \quad \vec{b} = \overrightarrow{RP}, \quad \vec{c} = \overrightarrow{PQ} a = QR , b = R P , c = P Q a ⃗ + b ⃗ + c ⃗ = 0 ⃗ \vec{a} + \vec{b} + \vec{c} = \vec{0} a + b + c = 0 Plan: 1. a ⃗ + b ⃗ + c ⃗ = 0 ⃗ ⇒ a ⃗ ⋅ b ⃗ \text{Plan: } 1.\ \vec{a} + \vec{b} + \vec{c} = \vec{0} \Rightarrow \vec{a} \cdot \vec{b} Plan: 1. a + b + c = 0 ⇒ a ⋅ b 2. ∣ c ⃗ ∣ 2 = ∣ a ⃗ + b ⃗ ∣ 2 2.\ |\vec{c}|^{2} = |\vec{a} + \vec{b}|^{2} 2. ∣ c ∣ 2 = ∣ a + b ∣ 2 3. c ⃗ × a ⃗ = a ⃗ × b ⃗ 3.\ \vec{c} \times \vec{a} = \vec{a} \times \vec{b} 3. c × a = a × b ⇒ a ⃗ ⋅ b ⃗ + b ⃗ ⋅ b ⃗ + b ⃗ ⋅ c ⃗ = 0 \Rightarrow \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{b} + \vec{b} \cdot \vec{c} = 0 ⇒ a ⋅ b + b ⋅ b + b ⋅ c = 0 a ⃗ ⋅ b ⃗ + 48 + 24 = 0 \vec{a} \cdot \vec{b} + 48 + 24 = 0 a ⋅ b + 48 + 24 = 0 a ⃗ ⋅ b ⃗ = − 72 ⇒ option (4) is true \vec{a} \cdot \vec{b} = -72 \quad \Rightarrow \ \text{option (4) is true} a ⋅ b = − 72 ⇒ option (4) is true c ⃗ = − ( a ⃗ + b ⃗ ) ⇒ ∣ c ⃗ ∣ 2 = ∣ a ⃗ + b ⃗ ∣ 2 \vec{c} = -(\vec{a} + \vec{b}) \Rightarrow |\vec{c}|^{2} = |\vec{a} + \vec{b}|^{2} c = − ( a + b ) ⇒ ∣ c ∣ 2 = ∣ a + b ∣ 2 = ∣ a ⃗ ∣ 2 + ∣ b ⃗ ∣ 2 + 2 a ⃗ ⋅ b ⃗ = |\vec{a}|^{2} + |\vec{b}|^{2} + 2\vec{a} \cdot \vec{b} = ∣ a ∣ 2 + ∣ b ∣ 2 + 2 a ⋅ b ∣ c ⃗ ∣ 2 = 144 + 48 − 144 = 48 |\vec{c}|^{2} = 144 + 48 - 144 = 48 ∣ c ∣ 2 = 144 + 48 − 144 = 48 ∣ c ⃗ ∣ 2 2 − ∣ a ⃗ ∣ = 48 2 − 12 \dfrac{|\vec{c}|^{2}}{2} - |\vec{a}| = \dfrac{48}{2} - 12 2 ∣ c ∣ 2 − ∣ a ∣ = 2 48 − 12 = 24 − 12 = 12 ✓ = 24 - 12 = 12 \quad \checkmark = 24 − 12 = 12 ✓ ∣ c ⃗ ∣ 2 2 + ∣ a ⃗ ∣ = 24 + 12 = 36 \dfrac{|\vec{c}|^{2}}{2} + |\vec{a}| = 24 + 12 = 36 2 ∣ c ∣ 2 + ∣ a ∣ = 24 + 12 = 36 36 ≠ 30 so option (2) is false 36 \neq 30 \quad \text{so option (2) is false} 36 = 30 so option (2) is false c ⃗ × a ⃗ = − ( a ⃗ + b ⃗ ) × a ⃗ \vec{c} \times \vec{a} = -(\vec{a} + \vec{b}) \times \vec{a} c × a = − ( a + b ) × a = − ( a ⃗ × a ⃗ + b ⃗ × a ⃗ ) = a ⃗ × b ⃗ = -(\vec{a} \times \vec{a} + \vec{b} \times \vec{a}) = \vec{a} \times \vec{b} = − ( a × a + b × a ) = a × b a ⃗ × b ⃗ + c ⃗ × a ⃗ = 2 ( a ⃗ × b ⃗ ) \vec{a} \times \vec{b} + \vec{c} \times \vec{a} = 2(\vec{a} \times \vec{b}) a × b + c × a = 2 ( a × b ) ∣ a ⃗ × b ⃗ + c ⃗ × a ⃗ ∣ = 2 ∣ a ⃗ × b ⃗ ∣ |\vec{a} \times \vec{b} + \vec{c} \times \vec{a}| = 2|\vec{a} \times \vec{b}| ∣ a × b + c × a ∣ = 2∣ a × b ∣ ∣ a ⃗ × b ⃗ ∣ 2 = ∣ a ⃗ ∣ 2 ∣ b ⃗ ∣ 2 − ( a ⃗ ⋅ b ⃗ ) 2 |\vec{a} \times \vec{b}|^{2} = |\vec{a}|^{2}|\vec{b}|^{2} - (\vec{a} \cdot \vec{b})^{2} ∣ a × b ∣ 2 = ∣ a ∣ 2 ∣ b ∣ 2 − ( a ⋅ b ) 2 = 144 ⋅ 48 − ( − 72 ) 2 = 1728 = 144 \cdot 48 - (-72)^{2} = 1728 = 144 ⋅ 48 − ( − 72 ) 2 = 1728 ∣ a ⃗ × b ⃗ ∣ = 1728 = 24 3 |\vec{a} \times \vec{b}| = \sqrt{1728} = 24\sqrt{3} ∣ a × b ∣ = 1728 = 24 3 ∣ a ⃗ × b ⃗ + c ⃗ × a ⃗ ∣ = 2 ⋅ 24 3 = 48 3 ✓ |\vec{a} \times \vec{b} + \vec{c} \times \vec{a}| = 2 \cdot 24\sqrt{3} = 48\sqrt{3} \quad \checkmark ∣ a × b + c × a ∣ = 2 ⋅ 24 3 = 48 3 ✓ True options: ( 1 ) , ( 3 ) , ( 4 ) \text{True options: } (1),\ (3),\ (4) True options: ( 1 ) , ( 3 ) , ( 4 )
