2017 TJC MYE Q4

\(ABCD\) is a piece of cardboard in the shape of a trapezium with \(\angle COA=\angle BAO={{90}^{\circ }}\), \(OA=5\)cm, \(OC=\frac{5\sqrt{2}}{3}\)cm and \(AB=5\sqrt{2}\) cm.

As part of a model for a structure, this cardboard is placed on a flat floor with \(OA\) touching the floor and \(B\) being at a height of \(5\)cm vertically above the floor.

The flat floor is assumed to be the \(x-y\) plane and the positive \(z\)-axis in the vertically upwards direction.

The position of \(O\) on the ground is taken to be the origin and \(A\) has coordinates \(\left( 3,4,0 \right)\).

1. Show that a possible coordinates of \(B\) is \(\left( 7,1,5 \right)\).[5]

Use \(\left( 7,1,5 \right)\) as the coordinates of \(B\) for the rest of the question.

2. Find the position vector of \(C\).[2]
3. Find a vector equation of line \(BC\).[2]
4. Find the acute angle between \(BC\) and the floor.[3]