Referred to the origin $O$, points $A$ and $B$ have position vectors $\mathbf{a}$ and \(\mat

Question

Referred to the origin $O$, points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively, where $\mathbf{a}$ and $\mathbf{b}$ are non-zero and non-parallel vectors. Point $C$ is the midpoint of $OA$ and point $D$ lies on $BC$ such that $BD:DC=2:3$. Find the position vectors $\overrightarrow{OC}$ and $\overrightarrow{OD}$, giving your answers in terms of $\mathbf{a}$ and $\mathbf{b}$.[2] Find the area of triangle $OCD$ in the form of $k\left| \mathbf{a}\times \mathbf{b} \right|$, where $k$ is a constant to be found.[3] Given that $\left| \mathbf{a} \right|=2$, $\left| \mathbf{b} \right|=\sqrt{2}$ and $\angle AOB=\frac{\pi }{4}$ radian, using scalar product, show that $CD$ is perpendicular to $OA$.[3] Given instead that $\mathbf{a}$ is a unit vector, give a geometrical interpretation of $\left| \mathbf{a}\cdot \mathbf{b} \right|$.[1]

Accepted Answer

(i) $\overrightarrow{OC}=\frac12\mathbf a$, $\overrightarrow{OD}=\frac15\mathbf a+\frac35\mathbf b$. (ii) $\frac3{20}|\mathbf a\times\mathbf b|$, so $k=\frac3{20}$. (iii) $CD\perp OA$. (iv) The length of the orthogonal projection of $OB$ onto the line through $O$ parallel to $OA$.

2024 JPJC Promo Q11

2024 JPJC Promo Q11

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