2021 ACJC Promo Q10
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.
(i)
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 $B$.
(ii)
Find the shortest distance from point $A$ to the line $L$.
[2]
(iii)
Show that the cartesian equation of the plane $\pi $ is $x+y-z=5$.
[2]
(iv)
Find the position vector of the foot of the perpendicular from point $A$ to the plane $\pi $.
[3]
(v)
Hence find the reflection of the line that passes through points $A$ and $B$ about the plane $\pi $.
[2]
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