Given: ∣ x ⃗ ∣ = ∣ y ⃗ ∣ = ∣ z ⃗ ∣ = 2 \text{Given: } |\vec{x}| = |\vec{y}| = |\vec{z}| = \sqrt{2} Given: ∣ x ∣ = ∣ y ∣ = ∣ z ∣ = 2
angle between each pair = π 3 \text{angle between each pair} = \dfrac{\pi}{3} angle between each pair = 3 π
To find: which of (1)–(4) is true \text{To find: which of (1)--(4) is true} To find: which of (1)–(4) is true
a ⃗ ⊥ x ⃗ , a ⃗ ⊥ ( y ⃗ × z ⃗ ) \vec{a} \perp \vec{x},\quad \vec{a} \perp (\vec{y} \times \vec{z}) a ⊥ x , a ⊥ ( y × z )
b ⃗ ⊥ y ⃗ , b ⃗ ⊥ ( z ⃗ × x ⃗ ) \vec{b} \perp \vec{y},\quad \vec{b} \perp (\vec{z} \times \vec{x}) b ⊥ y , b ⊥ ( z × x )
x ⃗ ⋅ y ⃗ = ∣ x ⃗ ∣ ∣ y ⃗ ∣ cos π 3 = 2 ⋅ 2 ⋅ 1 2 = 1 \vec{x} \cdot \vec{y} = |\vec{x}||\vec{y}|\cos\dfrac{\pi}{3} = \sqrt{2}\cdot\sqrt{2}\cdot\dfrac{1}{2} = 1 x ⋅ y = ∣ x ∣∣ y ∣ cos 3 π = 2 ⋅ 2 ⋅ 2 1 = 1
x ⃗ ⋅ z ⃗ = y ⃗ ⋅ z ⃗ = 1 \vec{x} \cdot \vec{z} = \vec{y} \cdot \vec{z} = 1 x ⋅ z = y ⋅ z = 1
Plan: a ⃗ ∥ x ⃗ × ( y ⃗ × z ⃗ ) , b ⃗ ∥ y ⃗ × ( z ⃗ × x ⃗ ) \text{Plan: } \vec{a} \parallel \vec{x} \times (\vec{y} \times \vec{z}),\quad \vec{b} \parallel \vec{y} \times (\vec{z} \times \vec{x}) Plan: a ∥ x × ( y × z ) , b ∥ y × ( z × x )
find a ⃗ , b ⃗ as multiples of y ⃗ − z ⃗ , z ⃗ − x ⃗ \text{find } \vec{a}, \vec{b} \text{ as multiples of } \vec{y} - \vec{z},\ \vec{z} - \vec{x} find a , b as multiples of y − z , z − x
then check options (1)–(4) \text{then check options (1)--(4)} then check options (1)–(4)
x ⃗ × ( y ⃗ × z ⃗ ) = y ⃗ ( x ⃗ ⋅ z ⃗ ) − z ⃗ ( x ⃗ ⋅ y ⃗ ) \vec{x} \times (\vec{y} \times \vec{z}) = \vec{y}(\vec{x} \cdot \vec{z}) - \vec{z}(\vec{x} \cdot \vec{y}) x × ( y × z ) = y ( x ⋅ z ) − z ( x ⋅ y )
= y ⃗ ( 1 ) − z ⃗ ( 1 ) = y ⃗ − z ⃗ = \vec{y}(1) - \vec{z}(1) = \vec{y} - \vec{z} = y ( 1 ) − z ( 1 ) = y − z
∴ a ⃗ ∥ y ⃗ − z ⃗ , a ⃗ = λ ( y ⃗ − z ⃗ ) \therefore \vec{a} \parallel \vec{y} - \vec{z},\quad \vec{a} = \lambda(\vec{y} - \vec{z}) ∴ a ∥ y − z , a = λ ( y − z )
( y ⃗ − z ⃗ ) ⋅ x ⃗ = y ⃗ ⋅ x ⃗ − z ⃗ ⋅ x ⃗ = 1 − 1 = 0 (\vec{y} - \vec{z}) \cdot \vec{x} = \vec{y} \cdot \vec{x} - \vec{z} \cdot \vec{x} = 1 - 1 = 0 ( y − z ) ⋅ x = y ⋅ x − z ⋅ x = 1 − 1 = 0
∴ a ⃗ ⋅ x ⃗ = 0 ✓ \therefore \vec{a} \cdot \vec{x} = 0 \quad \checkmark ∴ a ⋅ x = 0 ✓
y ⃗ × ( z ⃗ × x ⃗ ) = z ⃗ ( y ⃗ ⋅ x ⃗ ) − x ⃗ ( y ⃗ ⋅ z ⃗ ) \vec{y} \times (\vec{z} \times \vec{x}) = \vec{z}(\vec{y} \cdot \vec{x}) - \vec{x}(\vec{y} \cdot \vec{z}) y × ( z × x ) = z ( y ⋅ x ) − x ( y ⋅ z )
= z ⃗ − x ⃗ = \vec{z} - \vec{x} = z − x
∴ b ⃗ ∥ z ⃗ − x ⃗ , b ⃗ = μ ( z ⃗ − x ⃗ ) \therefore \vec{b} \parallel \vec{z} - \vec{x},\quad \vec{b} = \mu(\vec{z} - \vec{x}) ∴ b ∥ z − x , b = μ ( z − x )
b ⃗ ⋅ z ⃗ = μ ( z ⃗ − x ⃗ ) ⋅ z ⃗ = μ ( ∣ z ⃗ ∣ 2 − x ⃗ ⋅ z ⃗ ) = μ ( 2 − 1 ) = μ \vec{b} \cdot \vec{z} = \mu(\vec{z} - \vec{x}) \cdot \vec{z} = \mu(|\vec{z}|^{2} - \vec{x} \cdot \vec{z}) = \mu(2 - 1) = \mu b ⋅ z = μ ( z − x ) ⋅ z = μ ( ∣ z ∣ 2 − x ⋅ z ) = μ ( 2 − 1 ) = μ
( b ⃗ ⋅ z ⃗ ) ( z ⃗ − x ⃗ ) = μ ( z ⃗ − x ⃗ ) = b ⃗ (\vec{b} \cdot \vec{z})(\vec{z} - \vec{x}) = \mu(\vec{z} - \vec{x}) = \vec{b} ( b ⋅ z ) ( z − x ) = μ ( z − x ) = b
Option (1) is TRUE ✓ \text{Option (1) is TRUE} \quad \checkmark Option (1) is TRUE ✓
a ⃗ ⋅ y ⃗ = λ ( y ⃗ − z ⃗ ) ⋅ y ⃗ = λ ( ∣ y ⃗ ∣ 2 − z ⃗ ⋅ y ⃗ ) = λ ( 2 − 1 ) = λ \vec{a} \cdot \vec{y} = \lambda(\vec{y} - \vec{z}) \cdot \vec{y} = \lambda(|\vec{y}|^{2} - \vec{z} \cdot \vec{y}) = \lambda(2 - 1) = \lambda a ⋅ y = λ ( y − z ) ⋅ y = λ ( ∣ y ∣ 2 − z ⋅ y ) = λ ( 2 − 1 ) = λ
( a ⃗ ⋅ y ⃗ ) ( y ⃗ − z ⃗ ) = λ ( y ⃗ − z ⃗ ) = a ⃗ (\vec{a} \cdot \vec{y})(\vec{y} - \vec{z}) = \lambda(\vec{y} - \vec{z}) = \vec{a} ( a ⋅ y ) ( y − z ) = λ ( y − z ) = a
Option (2) is TRUE ✓ \text{Option (2) is TRUE} \quad \checkmark Option (2) is TRUE ✓
( y ⃗ − z ⃗ ) ⋅ ( z ⃗ − x ⃗ ) = y ⃗ ⋅ z ⃗ − y ⃗ ⋅ x ⃗ − ∣ z ⃗ ∣ 2 + z ⃗ ⋅ x ⃗ (\vec{y} - \vec{z}) \cdot (\vec{z} - \vec{x}) = \vec{y} \cdot \vec{z} - \vec{y} \cdot \vec{x} - |\vec{z}|^{2} + \vec{z} \cdot \vec{x} ( y − z ) ⋅ ( z − x ) = y ⋅ z − y ⋅ x − ∣ z ∣ 2 + z ⋅ x
= 1 − 1 − 2 + 1 = − 1 = 1 - 1 - 2 + 1 = -1 = 1 − 1 − 2 + 1 = − 1
a ⃗ ⋅ b ⃗ = λ μ [ ( y ⃗ − z ⃗ ) ⋅ ( z ⃗ − x ⃗ ) ] = λ μ ( − 1 ) = − λ μ \vec{a} \cdot \vec{b} = \lambda\mu\,[(\vec{y} - \vec{z}) \cdot (\vec{z} - \vec{x})] = \lambda\mu(-1) = -\lambda\mu a ⋅ b = λ μ [( y − z ) ⋅ ( z − x )] = λ μ ( − 1 ) = − λ μ
( a ⃗ ⋅ y ⃗ ) ( b ⃗ ⋅ z ⃗ ) = λ μ ⇒ a ⃗ ⋅ b ⃗ = − ( a ⃗ ⋅ y ⃗ ) ( b ⃗ ⋅ z ⃗ ) (\vec{a} \cdot \vec{y})(\vec{b} \cdot \vec{z}) = \lambda\mu \Rightarrow \vec{a} \cdot \vec{b} = -(\vec{a} \cdot \vec{y})(\vec{b} \cdot \vec{z}) ( a ⋅ y ) ( b ⋅ z ) = λ μ ⇒ a ⋅ b = − ( a ⋅ y ) ( b ⋅ z )
Option (3) is TRUE ✓ \text{Option (3) is TRUE} \quad \checkmark Option (3) is TRUE ✓
− ( a ⃗ ⋅ y ⃗ ) ( z ⃗ − y ⃗ ) = − λ ( z ⃗ − y ⃗ ) = λ ( y ⃗ − z ⃗ ) = a ⃗ -(\vec{a} \cdot \vec{y})(\vec{z} - \vec{y}) = -\lambda(\vec{z} - \vec{y}) = \lambda(\vec{y} - \vec{z}) = \vec{a} − ( a ⋅ y ) ( z − y ) = − λ ( z − y ) = λ ( y − z ) = a
Option (4) is TRUE ✓ \text{Option (4) is TRUE} \quad \checkmark Option (4) is TRUE ✓
∴ Correct options: (1), (2), (3), (4) \therefore \text{Correct options: (1), (2), (3), (4)} ∴ Correct options: (1), (2), (3), (4)
