2026 NYJC P1 Q3

Referred to the origin \(O\), the points \(A\), \(B\) and \(C\) have non-zero position vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) respectively. It is given that \(\mathbf{a}\times\mathbf{b}=\mathbf{c}\) and \(\mathbf{b}\times\mathbf{c}=k\mathbf{a}\), where \(k\) is a positive constant.

1. Deduce that \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) are mutually perpendicular and find \(k\) in terms of \(|\mathbf{b}|\).[3]
2. It is now given that \(|\mathbf{a}|=1\) and \(|\mathbf{b}|=\sqrt3\). Using a suitable scalar product, find the exact value of \(|\mathbf{b}-\mathbf{c}|\).[3]
3. State the geometrical interpretation of \(|\mathbf{b}\times\mathbf{c}|\).[1]