Referred to the origin $O$, the points $A$, $B$ and $C$ have position vectors \(4\mathbf{i}-

Question

Referred to the origin $O$, the points $A$, $B$ and $C$ have position vectors $4\mathbf{i}-2\mathbf{j}$, $\alpha \mathbf{i}-\mathbf{j}+2\mathbf{k}$ and $-\mathbf{i}-7\mathbf{j}+\beta \mathbf{k}$ respectively, where $\alpha $ and $\beta $ are constants. Given that $A$, $B$ and $C$ are collinear, show that $\alpha =5$, and find the value of $\beta $.[3] The plane $\pi $ contains the line $L$, which has equation $\mathbf{r}=2\mathbf{i}+3\mathbf{j}+\mu \left( 2\mathbf{i}-\mathbf{j}+\mathbf{k} \right)$. The plane $\pi $ is also parallel to the line that passes through the points $A$ and $B$. Find the shortest distance from point $A$ to the line $L$.[2] Show that the cartesian equation of the plane $\pi $ is $x+y-z=5$.[2] Find the position vector of the foot of the perpendicular from point $A$ to the plane $\pi $.[3] Hence find the reflection of the line that passes through points $A$ and $B$ about the plane $\pi $.[2]

Accepted Answer

(i) $\alpha=5,\ \beta=-10$ (ii) $\displaystyle \sqrt{\frac{31}{2}}\text{ units}$ (iii) $\pi:x+y-z=5$ (iv) $\displaystyle \overrightarrow{OF}=\begin{pmatrix}5\ -1\ -1\end{pmatrix}$ (v) $\displaystyle \mathbf{r}=\begin{pmatrix}6\ 0\ -2\end{pmatrix}+\lambda\begin{pmatrix}1\ 1\ 2\end{pmatrix}$, $\lambda\in\mathbb{R}$

2021 ACJC Promo Q10

2021 ACJC Promo Q10

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