Referred to the origin $O$, the points $A$ and $B$ have position vectors $\overrightarrow{OA}=\mathb

Question

Referred to the origin $O$, the points $A$ and $B$ have position vectors $\overrightarrow{OA}=\mathbf{i}+\mathbf{j}+4\mathbf{k}$ and $\overrightarrow{OB}=4\mathbf{i}-2\mathbf{j}+13\mathbf{k}$ respectively. The line $l$ has equation $\mathbf{r}=\mathbf{i}-\mathbf{j}-2\mathbf{k}+\ \lambda \left( -\mathbf{i}+2\mathbf{j} \right),\lambda \in \mathbb{R}$. The line $AB$ passes through the points $A$ and $B$ and intersects the line $l$ at point $M$. Find $F$, the foot of perpendicular from point $B$ to the line $l$. [3] Find $\overrightarrow{OM}$, and hence find the vector equation of the line $A'B'$ where $A'$ and $B'$ are the points of reflection of $A$ and $B$ in the line $l$ respectively. [6] $P$ is the foot of perpendicular from point $A$ to the line $l$. Find the exact value of $\frac{\mathrm{Area of }\Delta APM}{\mathrm{Area of }\Delta BB'M}$. [4] Find the length of projection of $\overrightarrow{B'M}$ onto the line $AB$. [2]

Accepted Answer

(i) $\overrightarrow{OF} = \begin{pmatrix} 0 \\ -3 \\ -2 \end{pmatrix}$ (ii) $l_{B'M}: \mathbf{r} = \begin{pmatrix} 0 \\ -4 \\ -17 \end{pmatrix} + \alpha\begin{pmatrix} -1 \\ 7 \\ 15 \end{pmatrix}$ (iii) $\frac{2}{25}$ (iv) $\frac{37}{\sqrt{11}}$ units

2017 AJC MYE PI Q10

2017 AJC MYE PI Q10

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