1 A glass ornament OABCDEFG is a truncated pyramid on a rectangular base (see Fig. 7). All dimensions are in centimetres.
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\caption{Fig. 7}
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- Write down the vectors \(\overrightarrow { \mathrm { CD } }\) and \(\overrightarrow { \mathrm { CB } }\).
- Find the length of the edge CD.
- Show that the vector \(4 \mathbf { i } + \mathbf { k }\) is perpendicular to the vectors \(\overrightarrow { \mathrm { CD } }\) and \(\overrightarrow { \mathrm { CB } }\). Hence find the cartesian equation of the plane BCDE.
- Write down vector equations for the lines OG and AF .
Show that they meet at the point P with coordinates (5, 10, 40).
You may assume that the lines CD and BE also meet at the point P .
The volume of a pyramid is \(\frac { 1 } { 3 } \times\) area of base × height. - Find the volumes of the pyramids POABC and PDEFG .
Hence find the volume of the ornament.