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.
197 exam-style questions on OCR A Level Maths 1.10 Vectors, covering 1.10.1 Vectors in two dimensions, 1.10.2 Vectors in three dimensions (A-level only), 1.10.3 Magnitude and direction of vectors, 1.10.4 Basic operations on vectors, 1.10.5 Position vectors, 1.10.6 Distance between points, 1.10.7 Problem solving using vectors, 1.10.8 Vectors in kinematics, and 1.10 Vectors. Each one has a worked solution and a mark scheme showing where the marks go.