Given: ∣ a ⃗ ∣ = ∣ b ⃗ ∣ = ∣ c ⃗ ∣ = 1 \text{Given: } |\vec{a}| = |\vec{b}| = |\vec{c}| = 1 Given: ∣ a ∣ = ∣ b ∣ = ∣ c ∣ = 1
a ⃗ , b ⃗ , c ⃗ are non-coplanar \vec{a},\ \vec{b},\ \vec{c} \text{ are non-coplanar} a , b , c are non-coplanar
∠ ( a ⃗ , b ⃗ ) = ∠ ( b ⃗ , c ⃗ ) = ∠ ( c ⃗ , a ⃗ ) = π 3 \angle(\vec{a},\vec{b}) = \angle(\vec{b},\vec{c}) = \angle(\vec{c},\vec{a}) = \frac{\pi}{3} ∠ ( a , b ) = ∠ ( b , c ) = ∠ ( c , a ) = 3 π
a ⃗ ⋅ b ⃗ = b ⃗ ⋅ c ⃗ = c ⃗ ⋅ a ⃗ = cos π 3 = 1 2 \vec{a}\cdot\vec{b} = \vec{b}\cdot\vec{c} = \vec{c}\cdot\vec{a} = \cos\frac{\pi}{3} = \frac{1}{2} a ⋅ b = b ⋅ c = c ⋅ a = cos 3 π = 2 1
To find: p 2 + 2 q 2 + r 2 q 2 \text{To find: } \frac{p^{2} + 2q^{2} + r^{2}}{q^{2}} To find: q 2 p 2 + 2 q 2 + r 2
Plan: dot both sides with a ⃗ , b ⃗ , c ⃗ one at a time \text{Plan: dot both sides with } \vec{a},\ \vec{b},\ \vec{c} \text{ one at a time} Plan: dot both sides with a , b , c one at a time
( a ⃗ × b ⃗ ) ⋅ a ⃗ = 0 , ( a ⃗ × b ⃗ ) ⋅ b ⃗ = 0 (\vec{a}\times\vec{b})\cdot\vec{a} = 0, \quad (\vec{a}\times\vec{b})\cdot\vec{b} = 0 ( a × b ) ⋅ a = 0 , ( a × b ) ⋅ b = 0
⇒ three scalar equations in p , q , r \Rightarrow \text{ three scalar equations in } p,\ q,\ r ⇒ three scalar equations in p , q , r
Δ = a ⃗ ⋅ ( b ⃗ × c ⃗ ) = [ a ⃗ , b ⃗ , c ⃗ ] ≠ 0 \Delta = \vec{a}\cdot(\vec{b}\times\vec{c}) = [\vec{a},\vec{b},\vec{c}] \ne 0 Δ = a ⋅ ( b × c ) = [ a , b , c ] = 0
(non-coplanar ⇒ Δ ≠ 0 ) \text{(non-coplanar } \Rightarrow \Delta \ne 0) (non-coplanar ⇒ Δ = 0 )
Dot with a ⃗ : 0 + Δ = p ( a ⃗ ⋅ a ⃗ ) + q ( a ⃗ ⋅ b ⃗ ) + r ( a ⃗ ⋅ c ⃗ ) \text{Dot with } \vec{a}: \quad 0 + \Delta = p(\vec{a}\cdot\vec{a}) + q(\vec{a}\cdot\vec{b}) + r(\vec{a}\cdot\vec{c}) Dot with a : 0 + Δ = p ( a ⋅ a ) + q ( a ⋅ b ) + r ( a ⋅ c )
Δ = p + q + r 2 ( 1 ) \Delta = p + \frac{q+r}{2} \qquad (1) Δ = p + 2 q + r ( 1 )
Dot with b ⃗ : p ( a ⃗ ⋅ b ⃗ ) + q ( b ⃗ ⋅ b ⃗ ) + r ( c ⃗ ⋅ b ⃗ ) = 0 \text{Dot with } \vec{b}: \quad p(\vec{a}\cdot\vec{b}) + q(\vec{b}\cdot\vec{b}) + r(\vec{c}\cdot\vec{b}) = 0 Dot with b : p ( a ⋅ b ) + q ( b ⋅ b ) + r ( c ⋅ b ) = 0
p 2 + q + r 2 = 0 ( 2 ) \frac{p}{2} + q + \frac{r}{2} = 0 \qquad (2) 2 p + q + 2 r = 0 ( 2 )
Dot with c ⃗ : Δ + 0 = p ( a ⃗ ⋅ c ⃗ ) + q ( b ⃗ ⋅ c ⃗ ) + r ( c ⃗ ⋅ c ⃗ ) \text{Dot with } \vec{c}: \quad \Delta + 0 = p(\vec{a}\cdot\vec{c}) + q(\vec{b}\cdot\vec{c}) + r(\vec{c}\cdot\vec{c}) Dot with c : Δ + 0 = p ( a ⋅ c ) + q ( b ⋅ c ) + r ( c ⋅ c )
Δ = p + q 2 + r ( 3 ) \Delta = \frac{p+q}{2} + r \qquad (3) Δ = 2 p + q + r ( 3 )
Δ = p + q + r 2 ( 1 ) \Delta = p + \frac{q+r}{2} \qquad (1) Δ = p + 2 q + r ( 1 )
p 2 + q + r 2 = 0 ( 2 ) \frac{p}{2} + q + \frac{r}{2} = 0 \qquad (2) 2 p + q + 2 r = 0 ( 2 )
Δ = p + q 2 + r ( 3 ) \Delta = \frac{p+q}{2} + r \qquad (3) Δ = 2 p + q + r ( 3 )
From ( 2 ) : q = − p + r 2 \text{From } (2): \quad q = -\frac{p+r}{2} From ( 2 ) : q = − 2 p + r
( 1 ) − ( 3 ) : 0 = ( p + q + r 2 ) − ( p + q 2 + r ) (1) - (3): \quad 0 = \left(p + \frac{q+r}{2}\right) - \left(\frac{p+q}{2} + r\right) ( 1 ) − ( 3 ) : 0 = ( p + 2 q + r ) − ( 2 p + q + r )
0 = p − r 2 ⇒ p = r 0 = \frac{p-r}{2} \quad \Rightarrow \quad p = r 0 = 2 p − r ⇒ p = r
q = − p + r 2 = − p + p 2 = − p q = -\frac{p+r}{2} = -\frac{p+p}{2} = -p q = − 2 p + r = − 2 p + p = − p
( 1 ) : Δ = p + − p + p 2 = p (1): \quad \Delta = p + \frac{-p+p}{2} = p ( 1 ) : Δ = p + 2 − p + p = p
⇒ p = r = Δ , q = − Δ \Rightarrow \ p = r = \Delta, \qquad q = -\Delta ⇒ p = r = Δ , q = − Δ
p 2 + 2 q 2 + r 2 q 2 = Δ 2 + 2 ( − Δ ) 2 + Δ 2 ( − Δ ) 2 \frac{p^{2} + 2q^{2} + r^{2}}{q^{2}} = \frac{\Delta^{2} + 2(-\Delta)^{2} + \Delta^{2}}{(-\Delta)^{2}} q 2 p 2 + 2 q 2 + r 2 = ( − Δ ) 2 Δ 2 + 2 ( − Δ ) 2 + Δ 2
= Δ 2 + 2 Δ 2 + Δ 2 Δ 2 = 4 Δ 2 Δ 2 = \frac{\Delta^{2} + 2\Delta^{2} + \Delta^{2}}{\Delta^{2}} = \frac{4\Delta^{2}}{\Delta^{2}} = Δ 2 Δ 2 + 2 Δ 2 + Δ 2 = Δ 2 4 Δ 2
4 Δ 2 Δ 2 = 4 \frac{4\Delta^{2}}{\Delta^{2}} = 4 Δ 2 4 Δ 2 = 4
