Question 2 12 marks Paper 1 Section A Medium No calculator Free
Maximum mark: 12
(a) For non-zero vectors \( \mathbf{a} \) and \( \mathbf{b} \), show that
(i) if \( \left|\mathbf{a}-\mathbf{b}\right| = \left|\mathbf{a}+\mathbf{b}\right| \), then \( \mathbf{a} \) and \( \mathbf{b} \) are perpendicular.[2]
(ii) \( \left|\mathbf{a}\times\mathbf{b}\right|^{2} = \left|\mathbf{a}\right|^{2}\left|\mathbf{b}\right|^{2} - \left(\mathbf{a}\cdot\mathbf{b}\right)^{2} \).[3]
(b) The points A, B and C have position vectors \( \mathbf{a} \), \( \mathbf{b} \) and \( \mathbf{c} \).
(i) Show that the area of triangle ABC is \( \dfrac{1}{2}\left|\mathbf{a}\times\mathbf{b} + \mathbf{b}\times\mathbf{c} + \mathbf{c}\times\mathbf{a}\right| \).[4]
(ii) Hence, show that the shortest distance from B to AC is \( \dfrac{\left|\mathbf{a}\times\mathbf{b} + \mathbf{b}\times\mathbf{c} + \mathbf{c}\times\mathbf{a}\right|}{\left|\mathbf{c}-\mathbf{a}\right|} \).[3]
Report an issue
A wrong mark, a missing diagram, or a step that does not follow.
Sign in to send a report for this question.