Mathematics

Problem

\mathbf{a}, \mathbf{b}

and

\mathbf{c}

are three vectors such that

\mathbf{a} \cdot \mathbf{a} = \mathbf{b} \cdot \mathbf{b} = \mathbf{c} \cdot \mathbf{c} = 3

and

|\mathbf{a} - \mathbf{b}|^2 + |\mathbf{b} - \mathbf{c}|^2 + |\mathbf{c} - \mathbf{a}|^2 = 27

, then a.

\mathbf{a}, \mathbf{b}

and

\mathbf{c}

are necessarily coplanar. b.

\mathbf{a}, \mathbf{b}

and

\mathbf{c}

represent sides of a triangle in magnitude and direction c.

\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c} + \mathbf{c} \cdot \mathbf{a}

has the least value

-9/2

\mathbf{a}, \mathbf{b}

and

\mathbf{c}

represent orthogonal triad of vectors