2022 ACJC J2 MYE Q5
Timothy Gan
2022 ACJC J2 MYE Q5
The diagram shows a prism with triangular ends $ABF$ and $DCE$. $AD$, $BC$ and $FE$ are parallel edges of the prism. The rectangular plane $ADEF$ has equation $x-2y=0$ and the rectangular base $ABCD$ has equation $3x-y+z=4$. The coordinates of $C$ and $F$ are $\left( 1,0,1 \right)$ and $\left( 4,2,0 \right)$ respectively.
(i)
Find the acute angle between the plane $ADEF$ and the line $CF$.
[2]
(ii)
Find the length of projection of vector $\overrightarrow{CF}$ on the base $ABCD$.
[2]
(iii)
Find the position vector of the point $P$ on the line passing through $A$ and $D$, which is closest to the mid-point of line segment $CF$.
[4]
(iv)
Find the Cartesian equation of the third rectangular plane $BCEF$.
[2]
(v)
Find the distance between the two triangular ends of the prism.
[2]
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