ACJC/2021/JC1/Promo/P1/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 $\beta$.

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The plane $\pi$ contains the line $L$, which has equation $\mathbf{r}=2 \mathbf{i}+3 \mathbf{j}+\mu(2 \mathbf{i}-\mathbf{j}+\mathbf{k})$. 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 $L$.

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(iii) Show that the cartesian equation of the plane $\pi$ is $x+y-z=5$.

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(iv) Find the position vector of the foot of the perpendicular from point $A$ to the plane $\pi$.

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(v) Hence find the reflection of the line that passes through points $A$ and $B$ about the plane $\pi$.

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