[Source of question: Unknown]

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 Ox and Oy are horizontal, and Oz is vertical. The coordinates of triangle are , and , with unites in metres.

- Find the length of the side AC.
- Find the scalar product of AB⋅BC, and the angle BAC.
- Show 2i+3j+8k that is perpendicular to the lines AB and AC. Hence find the Cartesian equation of the plane ABC.
- 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.

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