Referred to an origin, $O$, the position vectors of two points $A$ and $B$ are $\mathbf{a}$ and $\ma

Question

Referred to an origin, $O$, the position vectors of two points $A$ and $B$ are $\mathbf{a}$ and $\mathbf{b}$ respectively. $\mathbf{a}$ and $\mathbf{b}$ are not parallel. The point $C$ lies on $AB$ produced such that $AB:BC$ is $1:3$. Find the position vector of $C$. [2] The point $D$ lies on $OB$ produced such that $OB:OD$ is $1:k$. Given also that $CD$ is perpendicular to $AB$, show that $k=\frac{4{{\left| \mathbf{b} \right|}^{2}}-7\mathbf{a}\cdot \mathbf{b}+3{{\left| \mathbf{a} \right|}^{2}}}{{{\left| \mathbf{b} \right|}^{2}}-\mathbf{a}\cdot \mathbf{b}}$. [3] Show that $OADC$ cannot be a parallelogram. [2] Find the area of triangle $ABD$ in terms of $k$. [3]

Accepted Answer

(i) $\mathbf c=4\mathbf b-3\mathbf a$ (ii) $k=\dfrac{4\lvert\mathbf b\rvert^2-7\mathbf a\cdot\mathbf b+3\lvert\mathbf a\rvert^2}{\lvert\mathbf b\rvert^2-\mathbf a\cdot\mathbf b}$ (iii) $OADC$ is not a parallelogram (iv) $\dfrac{k-1}{2}\lvert\mathbf a\times\mathbf b\rvert$ square units

2013 RVHS P1 Q9

2013 RVHS P1 Q9

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