A research satellite's trajectory l1 l_1\,l1 is modeled relative to a fixed origin O O\,O by the equation
r=(1−22)+λ(22−1) \mathbf{r} = \begin{pmatrix} 1 \\ -2 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 2 \\ -1 \end{pmatrix} r=1−22+λ22−1where λ \lambda\,λ is a scalar parameter. A tracking station at the origin O O\,O detects the satellite at point A A\,A on l1 l_1\,l1 when its distance from O O\,O is 29 \sqrt{29}\,29 units.
Show that at AAA, the parameter λ \lambda\,λ satisfies
9λ2−8λ−20=0 9\lambda^2 - 8\lambda - 20 = 0 9λ2−8λ−20=0(i) Show that one possible position vector for A A\,A is 5i+2j5\mathbf{i} + 2\mathbf{j}5i+2j.
(ii) Find the other possible position vector for AAA.
The signal beam l2 l_2\,l2 from the station is parallel to l1 l_1\,l1 and passes through OOO. A maintenance probe B B\,B lies on l2l_2l2.
Given that:
Find the area of triangle OABOABOAB, giving your answer to one decimal place.