Consider any three-dimensional vectors \(\underset{\sim}{a}=\overrightarrow{O A}, \underset{\sim}{b}=\overrightarrow{O B}\) and \(\underset{\sim}{c}=\overrightarrow{O C}\) that satisfy these three conditions
\(\underset{\sim}{a} \cdot \underset{\sim}{b}=1\)
\(\underset{\sim}{b} \cdot \underset{\sim}{c}=2\)
\(\underset{\sim}{c} \cdot \underset{\sim}{a}=3\).
Which of the following statements about the vectors is true?
- Two of \(\underset{\sim}{a}, \underset{\sim}{b}\) and \(\underset{\sim}{c}\) could be unit vectors.
- The points \(A, B\) and \(C\) could lie on a sphere centred at \(O\).
- For any three-dimensional vector \(\underset{\sim}{a}\), vectors \(\underset{\sim}{b}\) and \(\underset{\sim}{c}\) can be found so that \(\underset{\sim}{a}, \underset{\sim}{b}\) and \(\underset{\sim}{c}\) satisfy these three conditions.
- \(\forall \ \underset{\sim}{a}, \underset{\sim}{b}\) and \(\underset{\sim}{c}\) satisfying the conditions, \(\exists \ r, s\) and \(t\) such that \(r, s\) and \(t\) are positive real numbers and \(r\underset{\sim}{a}+s \underset{\sim}{b}+t \underset{\sim}{c}=\underset{\sim}{0}\).