<p>Relative to the origin $O$, the points $A$, $B$ and $C$ have position vectors given by $\overrightarrow{OA}=\mathbf{a}$, $\overrightarrow{OB}=\mathbf{b}$, $\overrightarrow{OC}=\mathbf{c}$ and $\mu \mathbf{a}=\left( \mu +1 \right)\mathbf{b}-\mathbf{c}$, where&nbsp;$\mu \in {{\mathbb{R}}^{+}}$ .</p>
<p>It is given that&nbsp;$OAB$ is an equilateral triangle.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that&nbsp;$\mathbf{a}\cdot \mathbf{b}=\frac{1}{2}{{\left| \mathbf{b} \right|}^{2}}$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">The point $D$ lies on $OB$ produced such that&nbsp;$OD=\lambda OB$ and $OD$ is perpendicular to $CD$. Find the value of $\lambda $ in terms of $\mu $.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Hence by using the vector product, find the area of $\Delta OCD$, leaving your answer in the form of $k{{\left| \mathbf{b} \right|}^{2}}$, where&nbsp;$k$ is to be expressed in terms of $\mu $.<span class="inline-marks">[4]</span></div></li></ol>

Relative to the origin $O$, the points $A$, $B$ and $C$ have position vectors given by $\overrightarrow{OA}=\mathbf{a}$, $\overrightarrow{OB}=\mathbf{b}$, $\overrightarrow{OC}=\mathbf{c}$ and $\mu \mathbf{a}=\left( \mu +1 \right)\mathbf{b}-\mathbf{c}$, where $\mu \in {{\mathbb{R}}^{+}}$ .

It is given that $OAB$ is an equilateral triangle.

Show that $\mathbf{a}\cdot \mathbf{b}=\frac{1}{2}{{\left| \mathbf{b} \right|}^{2}}$.

The point $D$ lies on $OB$ produced such that $OD=\lambda OB$ and $OD$ is perpendicular to $CD$. Find the value of $\lambda $ in terms of $\mu $.

Hence by using the vector product, find the area of $\Delta OCD$, leaving your answer in the form of $k{{\left| \mathbf{b} \right|}^{2}}$, where $k$ is to be expressed in terms of $\mu $.

2016 NYJC Promo Q9

2016 NYJC Promo Q9

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