As part of a sculpture, an artist erects a flat triangular sheet $ABC$ in his garden. The vertices

Question

As part of a sculpture, an artist erects a flat triangular sheet $ABC$ in his garden. The vertices are attached to vertical poles $DA$, $EB$ and $FC$. The coordinate axes ${{O}_{x}}$ and ${{O}_{y}}$ are horizontal, and ${{O}_{z}}$ is vertical. The coordinates of triangle are $A(2,0,2)$, $B(-2,0,1)$ and $C(0,4,3)$, with unites in metres. Find the length of the side $AC$.[2] Find the scalar product of $\overrightarrow{AB}\cdot \overrightarrow{AC}$, and the angle $BAC$.[4] Show that $2\mathbf{i}+3\mathbf{j}-8\mathbf{k}$is perpendicular to the lines $AB$ and $AC$. Hence find the Cartesian equation of the plane $ABC$.[4] The artist decides to erect another vertical pole $GH$ based at the point $G(1,1,0)$. Calculate the height of the pole if $H$ is to lie in the plane $ABC$.[5]

Accepted Answer

(i) $\sqrt{21}\text{ m}$ (ii) $\overrightarrow{AB}\mathbin{\cdot}\overrightarrow{AC}=7$, $\angle BAC=68.3^\circ$ (iii) $2x+3y-8z=-12$ (iv) $\dfrac{17}{8}\text{ m}$

2007 TPJC P2 Q4

2007 TPJC P2 Q4

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