HCI/1/6

A group of boys want to set up a camping tent. They lay down a rectangular tarp OABC on the horizontal ground with OA = 3 m and AB = 1.5 m and secure the points D and E vertically above O and B respectively, such that .

Assume that the tent takes the shape as shown above with 6 triangular surfaces and a rectangular base. The point O is taken as the origin and the unit vectors i, j and k are taken to be in the direction of , and respectively.

(i) Show that the line DE can be expressed as $r = 2k+\lambda(2i+j), \lambda \in \mathbb{R}$.

(ii) Find the Cartesian equation of the plane ADE.

(iii) Determine the acute angle between the planes ADE and OABC. Hence, or otherwise, find the acute angle between the planes ADE and CDE.

Note: Question can be made harder and trickier should Origin, O be placed in the center of the base OACB.