With respect to a fixed origin OOO, a research drone is monitoring a straight-line path l1l_1l1 given by the equation
r=(−325)+μ(21−2) \mathbf{r} = \begin{pmatrix} -3 \\ 2 \\ 5 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix} r=−325+μ21−2where μ\muμ is a scalar parameter.
The point AAA represents a specific checkpoint on l1l_1l1. Given that ∣OA→∣=70|\overrightarrow{OA}| = \sqrt{70}∣OA∣=70.
Show that at AAA, the parameter μ\muμ satisfies
9μ2−28μ−32=0 9\mu^2 - 28\mu - 32 = 0 9μ2−28μ−32=0(i) Show that one possible position vector for AAA is 5i+6j−3k5\mathbf{i} + 6\mathbf{j} - 3\mathbf{k}5i+6j−3k. (ii) Find the other possible position vector for AAA.
The path l2l_2l2 is parallel to l1l_1l1 and passes through the origin OOO. Given that:
Find the area of triangle OABOABOAB, giving your answer to one decimal place.
