2021 ACJC P2 Q2
2021 ACJC P2 Q2
6 marks
Prelims
Relative to the origin \(O\), the points \(A\), \(B\), \(C\) have position vectors \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\), such that \(OACB\) is a quadrilateral. Let \(P\), \(Q\), \(R\) and \(S\) be the midpoints of the line segments \(OA\), \(AC\), \(CB\) and \(OB\) respectively.
- Show that \(PQRS\) is a parallelogram.[2]
- For any vectors \(\mathbf{p}\) and \(\mathbf{q}\), state the condition for \(\left| \mathbf{p}+\mathbf{q} \right|=\left| \mathbf{p} \right|+\left| \mathbf{q} \right|\).[1]
- Hence, by considering vector products, show that the area of \(OACB\) is twice the area of \(PQRS\).[3]
Video Solution:
Video solution locked
A free account saves your progress; this video needs tuition enrolment.
Solution:
Solution locked
680+ questions here come with free worked solutions. A free account saves your questions and progress; this one needs tuition enrolment.
Create a free account to practise similar questions.
Answer:(ii) \({\left\vert{} \mathbf{p} + \mathbf{q} \right\vert{} = \left\vert{} \mathbf{p} \right\vert{} + \left\vert{} \mathbf{q} \right\vert{}}\) if and only if \({\mathbf{p}}\) and \({\mathbf{q}}\) are parallel and in the same direction.
(iii) \({\frac{1}{2} \left\vert{} \mathbf{a} \times \mathbf{c} \right\vert{} + \frac{1}{2} \left\vert{} \mathbf{b} \times \mathbf{c} \right\vert{} = 2}\) (Area of \({PQRS}\))
Need help? Join our JC Math tuition classes.
Learn more