If α⃗, β⃗, γ⃗ be the unit vectors satisfying the condition α⃗+β⃗+γ⃗=0, then show that α⃗.β⃗+β⃗.γ⃗+γ⃗.α⃗ = -3/2. Hence examine whether the vector γ⃗ can be orthogonal to the vectors α⃗ and β⃗. (3+2)

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> [!TLDR]
> Square $|\vec\alpha+\vec\beta+\vec\gamma|=0$ to get the dot-product identity; then test the orthogonality claim against it.
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> **Part 1 (3 marks):** Since $\vec\alpha+\vec\beta+\vec\gamma=\vec0$, its magnitude squared is $0$:
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> $$|\vec\alpha+\vec\beta+\vec\gamma|^2 = |\vec\alpha|^2+|\vec\beta|^2+|\vec\gamma|^2+2(\vec\alpha\cdot\vec\beta+\vec\beta\cdot\vec\gamma+\vec\gamma\cdot\vec\alpha) = 0$$
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> Since $\vec\alpha,\vec\beta,\vec\gamma$ are unit vectors, $|\vec\alpha|^2=|\vec\beta|^2=|\vec\gamma|^2=1$:
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> $$3+2(\vec\alpha\cdot\vec\beta+\vec\beta\cdot\vec\gamma+\vec\gamma\cdot\vec\alpha)=0 \;\Rightarrow\; \vec\alpha\cdot\vec\beta+\vec\beta\cdot\vec\gamma+\vec\gamma\cdot\vec\alpha = -\frac32$$
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> **Part 2 (2 marks):** Suppose $\vec\gamma$ WERE orthogonal to both $\vec\alpha$ and $\vec\beta$, i.e. $\vec\gamma\cdot\vec\alpha=0$ and $\vec\beta\cdot\vec\gamma=0$. Substituting into the identity just proved:
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> $$\vec\alpha\cdot\vec\beta+0+0=-\frac32 \;\Rightarrow\; \vec\alpha\cdot\vec\beta=-\frac32$$
> …

Q.If α⃗, β⃗, γ⃗ be the unit vectors satisfying the condition α⃗+β⃗+γ⃗=0, then show that α⃗.β⃗+β⃗.γ⃗+γ⃗.α⃗ = -3/2. Hence examine whether the vector γ⃗ can be orthogonal to the vectors α⃗ and β⃗. (3+2)

Concept understanding — Scalar (Dot) Product

Q.If α⃗, β⃗, γ⃗ be the unit vectors satisfying the condition α⃗+β⃗+γ⃗=0, then show that α⃗.β⃗+β⃗.γ⃗+γ⃗.α⃗ = -3/2. Hence examine whether the vector γ⃗ can be orthogonal to the vectors α⃗ and β⃗. (3+2)

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