Mathematics

The key is to expand each squared magnitude using the dot product: $|\mathbf{u} - \mathbf{v}|^2 = |\mathbf{u}|^2 - 2\mathbf{u} \cdot \mathbf{v} + |\mathbf{v}|^2$. Since all three vectors are unit vectors, each $|\mathbf{u}|^2 = 1$, so the sum becomes $6 - 2(\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c} + \mathbf{c} \cdot \mathbf{a})$. To find the maximum, you need to minimize the dot products—which happens when the vectors are positioned to make those products as negative as possible.

The key is to expand each squared magnitude using the dot product: $|\mathbf{u} - \mathbf{v}|^2 = |\mathbf{u}|^2 - 2\mathbf{u} \cdot \mathbf{v} + |\mathbf{v}|^2$. Since all three vectors are unit vectors, each $|\mathbf{u}|^2 = 1$, so the sum becomes $6 - 2(\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c} + \mathbf{c} \cdot \mathbf{a})$. To find the maximum, you need to minimize the dot products—which happens when the vectors are positioned to make those products as negative as possible.