# 2020 CJC Promo Q11

##### Timothy Gan

**2020 CJC Promo Q11**

A hanging ornament is constructed by connecting six metal rods to form a tetrahedron $VABC$ as shown in the diagram.

The points $\left( x,y,z \right)$ are defined relative to a reference point $O\left( 0,0,0 \right)$, where units are in centimetres. The coordinates of the points $A$, $B$, $C$ and $V$ are $\left( 1,2,0 \right)$, $\left( 6,3,0 \right)$, $\left( 2,7,0 \right)$ and $\left( 3,4,5 \right)$ respectively.

(i)

Find the equation of the plane $VBC$ in scalar-product form.

[3]

(ii)

Find the coordinates of the point $P$ on the plane $VBC$ which is nearest to the point $A$.

[3]

(iii)

Hence or otherwise, find the shortest distance from the point $A$ to the point $VBC$.

[2]

(iv)

Find the acute angle between the line $VA$ and the plane $VBC$.

[3]

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